# Riemann Solitons and Almost Riemann Solitons on Almost Kenmotsu Manifolds

V. Venkatesha  H. Aruna Kumara and Devaraja Mallesha Naik  Department of
Mathematics
Kuvempu University
Shankaraghatta
Karnataka 577 451
India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The aim of this article is to study the Riemann soliton and gradient almost
Riemann soliton on certain class of almost Kenmotsu manifolds. Also some
suitable examples of Kenmotsu and $(\kappa,\mu)^{\prime}$-almost Kenmotsu
manifolds are constructed to justify our results.

###### Key words and phrases:

Riemann soliton, Ricci soliton, almost Kenmotsu manifold, Einstein manifold.

###### 1991 Mathematics Subject Classification:

53C25, 53C15, 53D15

##  1\. Introduction

Hamilton [9] introduced the concept of Ricci flow. The idea of Ricci flow
generalized to the concept of Riemann flow (see [16, 17]). As an analog of
Ricci soliton, Hiric̆a and Udrişte [10] introduced and studied Riemann
soliton. This is appears as a self-similar solution of the Riemann flow [16]:

$\displaystyle\frac{\partial}{\partial t}G(t)=-2R(g(t)),\quad t\in[0,I],$

where $G=\frac{1}{2}g\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}g$,
$R$ is the Riemann curvature tensor associated to the metric $g$ and
$\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}$ is Kulkarni-Nomizu
product. These extensions are natural, since some results in the Riemann flow
resembles the case of Ricci flow (for detail, see [17]). For instance, the
Riemann flow satisfies the short time existence and uniqueness [11]. Further
the authors in [14] characterize the Riemann soliton in terms of infinitesimal
harmonic transformation. For $(0,2)$-tensors $A$ and $B$, thier Kulkarni-
Nomizu product $A\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}B$ is
given by

$\displaystyle(A\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}B)(X_{1},X_{2},X_{3},$
$\displaystyle
X_{4})=A(X_{1},X_{3})B(X_{2},X_{4})+A(X_{2},X_{4})B(X_{1},X_{3})$
$\displaystyle-A(X_{1},X_{4})B(X_{2},X_{3})-A(X_{2},X_{3})B(X_{1},X_{4}).$
(1.1)

Riemann soliton is a smooth manifold $M$ together with Riemannian metric $g$
that satisfies

$\displaystyle
2R+\lambda(g\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}g)+(g\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}\pounds_{V}g)=0,$
(1.2)

where $V$ is a vector field called as a potential vector field, $\pounds_{V}$
denotes the Lie-derivative and $\lambda$ is a constant. The Riemann soliton
also corresponds as a fixed point of the Riemann flow, and they can be viewed
as a dynamical system, on the space of Riemannian metric modulo
diffeomorphism. Note that notion of Riemann soliton generalizes the space of
constant sectional curvature (i.e.,
$R=kg\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}g$, for some
constant $k$). A Riemann soliton is called shrinking when $\lambda<0$, steady
when $\lambda=0$ and expanding $\lambda>0$. If the vector field $V$ is the
gradient of a potential function $u$, then we get the notion of gradient
Riemann soliton. In such a case, the equation (1.2) transform into

$\displaystyle
R+\frac{1}{2}\lambda(g\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}g)+(g\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}Hess~{}u)=0,$
(1.3)

where $Hess~{}u$ denotes the Hessian of the smooth function $u$. In [10], it
is proved that a Riemann soliton on a compact Riemannian manifold is a
gradient. A Riemann soliton is called trivial when potential vector field $V$
vanishes identically, and this case manifold is of constant sectional
curvature. If $\lambda$ appearing in equation (1.2) and (1.3) is a smooth
function, then $g$ is called almost Riemann soliton and almost gradient
Riemann soliton respectively.

In [10], Hiric̆a and Udrişte studied Riemann soliton and gradient Riemann
soliton within the framework of Sasakian manifold. In their study, it is
proved that if Sasakian manifold admits a Riemann soliton whose soliton vector
field $V$ is pointwise collinear with $\xi$ (or a gradient Riemann soliton and
potential function is harmonic), then it is Sasakian space form. In this
study, the authors exposes an open problem as classify gradient Riemann
soliton for arbitrary potential function on Sasakian manifold. To fulfill this
classification, the present authors in [4] consider a Riemann soliton on
contact manifold, and obtained several intersting results. These results on
contact geometry intrigues us to study the Riemann soliton on other almost
contact metric manifolds. In this paper, we classify certain class of almost
Kenmotsu manifold which admits a Riemann soliton and almost gradient Riemann
solion.

##  2\. Preliminaries

Let $(M,g)$ be a $(2n+1)$-dimensional smooth Riemannian manifold. On this
manifold if there exists a (1,1)-type tensor field $\varphi$, a global vector
field $\xi$ and 1-form $\eta$ such that

$\displaystyle\varphi^{2}X=-X+\eta(X)\xi,\quad\eta(\xi)=1,$ (2.1)
$\displaystyle g(\varphi X,\varphi Y)=g(X,Y)-\eta(X)\eta(Y),$ (2.2)

for any vector field $X,Y$ on $M$, then we say that $(\varphi,\xi,\eta,g)$ is
an almost contact metric structure and $M$ is an almost contact metric
manifold [1]. Generally, $\xi$ and $\eta$ are called Reeb or characterstic
vector field and almost contact 1-form respectively. We remark that an almost
contact metric structure on a Riemannian manifold $M$ may be regarded as a
reduction of the structure group $M$ to $U(n)\times 1$. For such a manifold,
we define a fundamental 2-form $\Phi$ by $\Phi(X,Y)=g(X,\varphi Y)$. It is
well known that the normality of an almost contact structure is expressed by
vanishing of the tensor $N_{\varphi}=[\varphi,\varphi]+2d\eta\otimes\xi$,
where $[\varphi,\varphi]$ is the Nijenhuis tensor of $\varphi$ (see [1]).

Almost Kenmotsu manifold [8] is an almost contact metric manifold such that
$d\eta=0$ and $d\Phi=2\eta\wedge\Phi$. A normal almost Kenmotsu manifold is
called Kenmotsu manifold, and this normality condition is expressed as

$\displaystyle(\nabla_{X}\varphi)Y=g(\varphi X,Y)\xi-\eta(Y)\varphi X,$

for any vector field $X,Y$ on $M$, where $\nabla$ denotes the Levi-Civita
connection of the Riemannian metric $g$. On an almost Kenmotsu manifold, we
set $\ell=R(\cdot,\xi)\xi$, $2h=\pounds_{\xi}\varphi$ and
$h^{\prime}=h\circ\varphi$, where $R$ denotes the curvature tensor of $M$ and
$\pounds$ is the Lie-derivative. These mentioned tensor fields plays important
role in almost Kenmotsu manifold, and it is easily seen that both are
symmetric and satisfies the following equations

$\displaystyle\nabla_{X}\xi=X-\eta(X)\xi+h^{\prime}X,$ (2.3) $\displaystyle
h\xi=\ell\xi=0,\quad trh=trh^{\prime}=0,\quad h\varphi+\varphi h=0,$ (2.4)
$\displaystyle tr\ell=S(\xi,\xi)=g(Q\xi,\xi)=-2n-trh^{2},$ (2.5)

where $S$ is the Ricci curvature tensor, $Q$ is the Ricci operator with
respect to $g$ anb $tr$ is the trace operator.

######  Definition 2.1.

On almost contact metric manifold $M$, a vector field $V$ is said to be
infinitesimal contact transformation if $\pounds_{V}\eta=\sigma\eta$, for some
function $\sigma$. In particular, we called $V$ is a strict infinitesimal
contact transformation if $\pounds_{V}\eta=0$.

Now, we recall some formulas which are helpfull to prove our main results.
Using the symmetry of $\pounds_{V}\nabla$ in the commutation formula (see Yano
[22]):

$\displaystyle(\pounds_{V}$
$\displaystyle\nabla_{X}g-\nabla_{X}\pounds_{V}g-\nabla_{[V,X]}g)(Y,Z)$
$\displaystyle=-g((\pounds_{V}\nabla)(X,Y),Z)-g((\pounds_{V}\nabla)(X,Z),Y),$

we derive

$\displaystyle
2g((\pounds_{V}\nabla)(X,Y),Z)=(\nabla_{X}\pounds_{V}g)(Y,Z)+(\nabla_{Y}\pounds_{V}g)(Z,X)-(\nabla_{Z}\pounds_{V}g)(X,Y).$
(2.6)

The following well-known commutation equations is also known to us from Yano
[22]:

$\displaystyle(\pounds_{V}R)(X,Y)Z=(\nabla_{X}\pounds_{V}\nabla)(Y,Z)-(\nabla_{Y}\pounds_{V}\nabla)(X,Z),$
(2.7)
$\displaystyle\pounds_{V}\nabla_{X}Y-\nabla_{X}\pounds_{V}Y-\nabla_{[V,X]}Y=(\pounds_{V}\nabla)(X,Y).$
(2.8)

##  3\. Riemann soliton and almost gradient Riemann soliton on Kenmotsu
manifolds

Dileo and Pastore [2] proved that the almost contact metric structure of an
almost Kenmotsu manifold is normal if and only if the foliations of the
distribution $\mathcal{D}$ are Kählerian and the $(1,1)$-type tensor field $h$
vanishes. In particular, we have immediately that almost Kenmotsu manifold is
a Kenmotsu manifold if and only if $h=0$. The following formulae are holds for
a Kenmotsu manifold [12]

$\displaystyle\nabla_{X}\xi=X-\eta(X)\xi,$ (3.1) $\displaystyle
R(X,Y)\xi=\eta(X)Y-\eta(Y)X,$ (3.2) $\displaystyle Q\xi=-2n\xi.$ (3.3)

Differentiating (3.3) along an arbitrary vector field $X$ and recalling (3.1)
we deduce

$\displaystyle(\nabla_{X}Q)\xi=-QX-2nX.$ (3.4)

In this section, we aim to investigate the existence of geometry of Riemann
soliton and an almost gradient Riemann soliton on Kenmotsu manifold. First, we
prove the following result.

######  Lemma 3.1.

Let $M$ be a $(2n+1)$-dimensional Kenmotsu manifold. If $(g,V)$ is a Riemann
soliton with soliton vector $V$ has a constant divergence, then the potential
vector field and Ricci tensor satisfy

$\displaystyle divV=2n(1-\lambda),$ (3.5)
$\displaystyle(\pounds_{V}S)(X,\xi)=\frac{1}{2n-1}\\{-(Yr)+(\xi r)\eta(Y)\\},$
(3.6)

where $div$ denotes the divergence operator.

###### Proof.

As a result of (1), the Riemann soliton equation (1.2) can be expressed as

$\displaystyle 2R(X,Y,$ $\displaystyle
Z,W)+2\lambda\\{g(X,W)g(Y,Z)-g(X,Z)g(Y,W)\\}$
$\displaystyle+\\{g(X,W)(\pounds_{V}g)(Y,Z)+g(Y,Z)(\pounds_{V}g)(X,W)$
$\displaystyle-g(X,Z)(\pounds_{V}g)(Y,W)-g(Y,W)(\pounds_{V}g)(X,Z)\\}=0.$
(3.7)

Contractiong (3.7) over $X$ and $W$, we obtain

$\displaystyle(\pounds_{V}g)(Y,Z)+\frac{2}{2n-1}S(Y,Z)+\frac{2}{2n-1}(2n\lambda+divV)g(Y,Z)=0.$
(3.8)

Taking covariant derivative of the above relation gives

$\displaystyle(\nabla_{X}\pounds_{V}g)(Y,Z)+\frac{2}{2n-1}(\nabla_{X}S)(Y,Z)=0,$
(3.9)

where we applied $V$ has a constant divergence. Inserting the foregoing
equation into (2.6) we obtain

$\displaystyle
g((\pounds_{V}\nabla)(X,Y),Z)=\frac{1}{2n-1}\\{(\nabla_{Z}S)(X,Y)-(\nabla_{X}S)(Y,Z)-(\nabla_{Y}S)(Z,X)\\}.$
(3.10)

The present authors in [18] and Ghosh in [7] independently gave the complete
proof of the following formula which is valid for any Kenmotsu manifold

$\displaystyle(\nabla_{\xi}Q)X=-2QX-4nX.$ (3.11)

Replacing $Y$ by $\xi$ in (3.10), applying (3.4) and (3.11) we get

$\displaystyle(\pounds_{V}\nabla)(X,\xi)=\frac{2}{2n-1}QX+\frac{4n}{2n-1}X.$
(3.12)

Differentiating (3.12) covariantly along $Y$ and utilizing (3.1), we find

$\displaystyle(\nabla_{Y}\pounds_{V}\nabla)(X,\xi)+(\pounds_{V}\nabla)(X,Y)-\frac{2}{2n-1}\eta(Y)\\{QX+2nX\\}=\frac{2}{2n-1}(\nabla_{Y}Q)X.$

Making use of this in the well known expression (2.7), we have

$\displaystyle(\pounds_{V}R)(X,Y)\xi=$
$\displaystyle\frac{2}{2n-1}\\{\eta(X)QY-\eta(Y)QX+(\nabla_{X}Q)Y-(\nabla_{Y}Q)X\\}$
$\displaystyle+\frac{4n}{2n-1}\\{\eta(X)Y-\eta(Y)X\\}.$ (3.13)

We insert $Y=\xi$ in the above relation and utilizing (3.4) and (3.11) to
achieve $(\pounds_{V}R)(X,\xi)\xi=0$. On the other hand, operating
$\pounds_{V}$ to the formula $R(X,\xi)\xi=-X+\eta(X)\xi$ (follows from (3.2))
yields

$\displaystyle(\pounds_{V}R)(X,\xi)\xi+g(X,\pounds_{V}\xi)\xi-2\eta(\pounds_{V}\xi)X=\\{(\pounds_{V}\eta)X\\}\xi,$

which by virtue of $(\pounds_{V}R)(X,\xi)\xi=0$ becomes

$\displaystyle
g(X,\pounds_{V}\xi)\xi-2\eta(\pounds_{V}\xi)X=\\{(\pounds_{V}\eta)X\\}\xi.$
(3.14)

With the aid of (3.3), the equation (3.8) takes the form

$\displaystyle(\pounds_{V}g)(X,\xi)=-\frac{2}{2n-1}\\{2n\lambda+divV-2n\\}\eta(X).$
(3.15)

Now, Lie-differentiating $\eta(X)=g(X,\xi)$ and $g(\xi,\xi)=1$ along $V$ and
taking into account (3.15) provides

$\displaystyle(\pounds_{V}\eta)X-g(X,\pounds_{V}\xi)+\frac{2}{2n-1}\\{2n\lambda+divV-2n\\}\eta(X)=0,$
$\displaystyle\eta(\pounds_{V}\xi)=\frac{1}{2n-1}\\{2n\lambda+divV-2n\\}.$

Utilizing these equations in (3.14) yields
$(2n\lambda+divV-2n)\\{X-\eta(X)\xi\\}=0$. Tracing this provides (3.5). Now
contracting the equation (3.13) provides

$\displaystyle(\pounds_{V}S)(Y,\xi)=\frac{1}{2n-1}\\{-(Yr)-2(r+2n(2n+1))\\}\eta(Y),$
(3.16)

where we applied the well-known formula $divQ=\frac{1}{2}grad\,\,r$ and
$tr\nabla Q=grad\,\,r$. Taking trace of (3.11) provides $(\xi
r)=-2(r+2n(2n+1))$. By virtue of this and (3.16) gives (3.6). This completes
the proof. ∎

######  Theorem 3.2.

Let $M$ be an $\eta$-Einstein Kenmotsu manifold of dimension higher than $3$.
If $(g,V)$ is a Riemann soliton with $divV=constant$, then $M$ is Einstein.

###### Proof.

We know that a Kenmotsu manifold $M$ is $\eta$-Einstein if and only if

$\displaystyle
S(X,Y)=\left(\frac{r}{2n}+1\right)g(X,Y)-\left(\frac{r}{2n}+2n+1\right)\eta(X)\eta(Y).$
(3.17)

With the aid of the above equation, it has been proved by present authors in
([18], Lemma 3.4) that on $\eta$-Einstein Kenmotsu manifold of $dim>3$ there
holds

$\displaystyle Dr=(\xi r)\xi.$

Making use of this in (3.6), we have $(\pounds_{V}S)(X,\xi)=0$. Now, applying
$\pounds_{V}$ to (3.3), recalling (3.17), (3.15), (3.5) and
$\eta(\pounds_{V}\xi)=0$, we obtain

$\displaystyle(r+2n(2n+1))\pounds_{V}\xi=0.$ (3.18)

Suppose that $r\neq 2n(2n+1)$ in some open set $\mathcal{O}$ of M. Then on
$\mathcal{O}$, $\pounds_{V}\xi=0=\pounds_{V}\eta$. Replacing $Y$ by $\xi$ in
(2.8) and using $\pounds_{V}\xi=0=\pounds_{V}\eta$, we have

$\displaystyle(\pounds_{V}\nabla)(X,\xi)=$
$\displaystyle\pounds_{V}X-\eta(X)\pounds_{V}\xi-\\{(\pounds_{V}\eta)X\\}\xi$
$\displaystyle-\eta(\pounds_{V}X)\xi-\pounds_{V}X+\eta(\pounds_{V}X)\xi=0.$

By virtue of this in (3.12), we have $QX=-2nX$. Taking trace of this gives
$r=-2n(2n+1)$ on $\mathcal{O}$, which is a contradiction on $\mathcal{O}$.
Thus, equation (3.18) gives $r=-2n(2n+1)$ and therefore we can conclude from
$\eta$-Einstein condition (3.17) that $M$ is Einstein. ∎

It is known that any $3$-dimensional Kenmotsu manifold is $\eta$-Einstein. As
a result of Theorem 3.2, it is interesting to study Riemann soliton in
Kenmotsu $3$-manifold and here, we prove the following outcome.

######  Theorem 3.3.

Let be a 3-dimensional Kenmotsu manifold. If $(g,V)$ represents a Riemann
soliton with soliton vector $V$ has a constant divergence, then $M$ is of
constant negative curvature $-1$.

###### Proof.

It is well known that the Riemannian curvature tensor of $3$-dimensional
Riemannian manifold is given by

$\displaystyle R(X,Y)Z=$ $\displaystyle g(Y,Z)QX-g(X,Z)QY+g(QY,Z)X-g(QX,Z)Y$
$\displaystyle-\frac{1}{2}\\{g(Y,Z)X-g(X,Z)Y\\}.$ (3.19)

Replacing $\xi$ in place of $Y$ and $Z$ in above equation, employing (3.2) and
(3.3) gives

$\displaystyle
QX=\left(\frac{r}{2}+1\right)X-\left(\frac{r}{2}+3\right)\eta(X)\xi,$

which is equivalent to

$\displaystyle
S(X,Y)=\left(\frac{r}{2}+1\right)g(X,Y)-\left(\frac{r}{2}+3\right)\eta(X)\eta(Y).$
(3.20)

Applying $\pounds_{V}$ to (3.3), recalling (3.20) yields

$\displaystyle(\pounds_{V}S)(X,\xi)+\left(\frac{r}{2}+1\right)g(X,\pounds_{V}\xi)-\left(\frac{r}{2}+3\right)\eta(X)\eta(\pounds_{V}\xi)=-2(\pounds_{V}\eta)(X).$

This together with (3.6) provides

$\displaystyle-Y(r)+\xi(r)\eta(Y)+\left(\frac{r}{2}+1\right)g(X,\pounds_{V}\xi)$
$\displaystyle-\left(\frac{r}{2}+3\right)\eta(X)\eta(\pounds_{V}\xi)+2(\pounds_{V}\eta)(X)=0.$
(3.21)

By virtue of (3.5) and (3.15), it follows from (3.21) that

$\displaystyle(r+6)g(X,\pounds_{V}\xi)=2\\{Y(r)-\xi(r)\eta(Y)\\},$ (3.22)

where we used Theorem 4.1 of Wang [20]. Now suppose that $r=-6$. Then from
(3.20) we have $QX=-2X$, and substituting this in (3.19) one gets

$\displaystyle R(X,Y)Z=-\\{g(Y,Z)X-g(X,Z)Y\\},$

which means $M$ is of constant curvature $-1$.

Now we suppose that $r\neq-6$ in some open set $\mathcal{O}$ of $M$. Then on
$\mathcal{O}$, the relation (3.22) can be written as

$\displaystyle\pounds_{V}\xi=f\\{Dr-\xi(r)\xi\\},$ (3.23)

where $f=\frac{2}{r+6}$. Taking $Y$ by $\xi$ in (2.8) and using (3.1), (3.23)
and (3.12), it follows that

$\displaystyle f\\{$ $\displaystyle
X(r)\eta(Y)+Y(r)\eta(X)-2\xi(r)\eta(X)\eta(Y)+g(\nabla_{X}Dr,Y)-X(\xi(r))\eta(Y)\\}$
$\displaystyle+X(f)\\{Y(r)-\xi(r)\eta(Y)\\}+\left\\{f~{}\xi(r)-\frac{1}{2}\xi(r)\right\\}g(\varphi
X,\varphi Y)=0.$

Antisymmetrizing the foregoing equation and keeeping in mind the Poincare
lemma: $g(\nabla_{X}Dr,Y)=g(\nabla_{Y}Dr,X)$, we find

$\displaystyle
f\\{Y(\xi(r))\eta(X)-X(\xi(r))\eta(Y)\\}+X(f)\\{Y(r)-\xi(r)\eta(Y)\\}$
$\displaystyle-Y(f)\\{X(r)-\xi(r)\eta(X)\\}=0.$

Putting $Y=\xi$ in the above equation and using Theorem 4.1 of Wang [20] one
get

$\displaystyle(2f-\xi(f))\\{X(r)-\xi(r)\eta(X)\\},$

which implies

$\displaystyle(2f-\xi(f))\\{Dr-\xi(r)\xi\\}.$ (3.24)

In the following first case we show that $Dr=\xi(r)\xi$, one get
$g(Dr,\xi)=-2(r+6)$. Since $f=\frac{2}{r+6}$ we obtain $\xi(f)=2f$. In the
second case we show $\xi(f)=2f$ implies $Dr=\xi(r)\xi$. Thus for any point
$p\in\mathcal{O}$, we have $(Dr)_{p}=\xi_{p}(r)\xi_{p}$ if and only if
$\xi_{p}(f)=2f$. Hence from (3.24), we have either $\xi(f)=2f$ or
$Dr=\xi(r)\xi$.

Case 1: First we assume

$\displaystyle Dr=\xi(r)\xi,$ (3.25)

and so (3.23) becomes $\pounds_{V}\xi=0$ on the open set $\mathcal{O}$ of $M$.
Setting $Y$ by $\xi$ in (2.8) and using $\pounds_{V}\xi=0$, we have
$(\pounds_{V}\nabla)(X,\xi)=0$. This together with (3.12) provides $QX=-2X$.
Contracting this with respect to $X$ gives $r=-6$ which yields a
contradiction.

Case 2: Suppose if $\xi(f)=2f$, then using $f=\frac{2}{r+6}$ we obtain
$\xi(r)=-2(r+6)$ which means

$\displaystyle g(Dr,\xi)=-2(r+6).$

As $r\neq-6$ on $\mathcal{O}$, the above equation implies $Dr=f\xi$, for some
smooth function $f$. In fact, we have $Dr=\xi(r)\xi$ and so following the Case
1 we arrive the contradiction. This completes the proof. ∎

Now, we consider a Kenmotsu metric as a gradient almost Riemann soliton and
first, we need the following result to prove our main theorem.

######  Lemma 3.4.

For gradient almost Riemann soliton, the following formula is valid

$\displaystyle R(X,Y)Du=$
$\displaystyle\frac{1}{2n-1}\\{(\nabla_{Y}Q)X-(\nabla_{X}Q)Y\\}$
$\displaystyle+\frac{1}{2n-1}\\{Y(2n\lambda+\Delta u)X-X(2n\lambda+\Delta
u)Y\\},$ (3.26)

where $\Delta u=divDu$, $\Delta$ is the Laplacian operator.

###### Proof.

Contracting gradient almost Riemann soliton equation (1.3), we get

$\displaystyle Hess~{}u+\frac{1}{2n-1}S+\frac{1}{2n-1}(2n\lambda+\Delta
u)g=0.$

Note that the above equation may exhibited as

$\displaystyle\nabla_{Y}Du=-\frac{1}{2n-1}QY-\frac{1}{2n-1}(2n\lambda+\Delta
u)Y.$ (3.27)

By straightforward computations, using the well known expression of the
curvature tensor:

$\displaystyle
R(X,Y)=\nabla_{X}\nabla_{Y}-\nabla_{Y}\nabla_{X}-\nabla_{[X,Y]},$

and the repeated use of equation (3.27) gives the desired result. ∎

######  Theorem 3.5.

If metric of a Kenmotsu manifold $M$ of dimeninsion $(2n+1)$ represents a
gradient almost Riemann soliton, then either $M$ is Einstein or the soliton
vector field $V$ is pointwise collinear with the characteristic vector field
$\xi$ on an open set $\mathcal{O}$ of $M$. In the first case, if $M$ is
complete, then $M$ is locally isometric to a hyperbolic space
$\mathbb{H}^{2n+1}$, and the function $2n\lambda+\Delta u$, upto an additive
constant, can be expressed as a linear combination of $cosh\,t$ and $sinh\,t$.

###### Proof.

Replacing $Y$ by $\xi$ in (3.26) and employing (3.4) and (3.11), we find

$\displaystyle
R(X,\xi)Du=-\frac{1}{2n-1}\\{QX+2nX\\}+\frac{1}{2n-1}\\{\xi(2n\lambda+\Delta
u)X-X(2n\lambda+\Delta u)\xi\\}.$ (3.28)

From (3.2), we have $R(X,\xi)Y=g(X,Y)\xi-\eta(Y)X$. Employing this into (3.28)
provides

$\displaystyle g(X,Du+\frac{1}{2n-1}D(2n\lambda+\Delta u))$
$\displaystyle\xi-\\{\xi(u)+\frac{1}{2n-1}\xi(2n\lambda+\Delta u)\\}X$
$\displaystyle=-\frac{1}{2n-1}\\{QX+2nX\\}.$ (3.29)

Inner product of the previous equation with $\xi$ and using (3.3) provides
$X(u+\frac{1}{2n-1}(2n\lambda+\Delta u))=\xi(u+\frac{1}{2n-1}(2n\lambda+\Delta
u))\eta(X)$, from which we have

$\displaystyle d(u+\frac{1}{2n-1}(2n\lambda+\Delta
u))=\xi(u+\frac{1}{2n-1}(2n\lambda+\Delta u))\eta,$ (3.30)

where $d$ is the exterior derivative. This means that
$u+\frac{1}{2n-1}(2n\lambda+\Delta u)$ is invariant along the distribution
$\mathcal{D}$, i.e., $X(u+\frac{1}{2n-1}(2n\lambda+\Delta u))=0$ for any
vector field $X\in\mathcal{D}$. Utilizing (3.30) into (3.29), we obtain

$\displaystyle\\{\xi(u)+\frac{1}{2n-1}\xi(2n\lambda+\Delta u)\\}\eta(X)$
$\displaystyle\xi-\\{\xi(u)+\frac{1}{2n-1}\xi(2n\lambda+\Delta u)\\}X$
$\displaystyle=-\frac{1}{2n-1}\\{QX+2nX\\}.$ (3.31)

Contracting the above equation, one immediately obtain

$\displaystyle\xi(u+\frac{1}{2n-1}(2n\lambda+\Delta
u))=\frac{1}{2n-1}\\{\frac{r}{2n}+2n+1\\}.$ (3.32)

Making use of last equation in (3.31) one can obtain $\eta$-Einstein condition
(3.17). Now, we contract equation (3.26) over $X$ to deduce

$\displaystyle S(Y,Du)=\frac{1}{2(2n-1)}Y(r)+\frac{2n}{2n-1}Y(2n\lambda+\Delta
u).$

In the contrast of above equation with (3.17) we obtain

$\displaystyle(r+2n)Y(u)-($ $\displaystyle
r+2n(2n+1))\xi(u)\eta(Y)-\frac{1}{2(2n-1)}Y(r)$
$\displaystyle-\frac{4n^{2}}{2n-1}Y(2n\lambda+\Delta u)=0.$ (3.33)

Inserting $Y=\xi$ in (3.33) and recalling (3.32) one can get
$\xi(r)=-2\\{r+2n(2n+1)\\}$. Action of $d$ on (3.30), we get $dr\wedge\eta=0$,
where we used $d^{2}=0$ and $d\eta=0$. Hence by virtue of
$\xi(r)=-2\\{r+2n(2n+1)\\}$ we have

$\displaystyle Dr=-2\\{r+2n(2n+1)\\}\xi.$ (3.34)

Suppose that $X$ in (3.33) is orthogonal to $\xi$. Taking into account
$(u+\frac{1}{2n-1}(2n\lambda+\Delta u))$ being a constant along $\mathcal{D}$
and utilizing (3.30) and (3.34), then we get $(r+2n(2n+1))X(u)=0$ for any
$X\in\mathcal{D}$. This implies that

$\displaystyle(r+2n(2n+1))(Du-\xi(u)\xi)=0.$ (3.35)

If $r=-2n(2n+1)$, then the equation (3.17) shows that $QX=-2nX$, and hence $M$
is Einstein. Since $r=-2n(2n+1)$, it follows from (3.32) that
$\xi(u)=-\frac{1}{2n-1}\xi(2n\lambda+\Delta u)$ and therefore
$Du=-\frac{1}{2n-1}D(2n\lambda+\Delta u)\xi$. Thus, equation (3.27) can be
exhibited as

$\displaystyle\nabla_{X}D(2n\lambda+\Delta u)=((2n\lambda+\Delta u)-2n)X.$
(3.36)

According to Theorem 2 of Tashiro [15] we obtain $M$ is locally isometric to
the hyperbolic space $\mathbb{H}^{2n+1}$, when $M$ is complete. Since
$\nabla_{\xi}\xi=0$ and $g(\xi,D(2n\lambda+\Delta u))=\xi(2n\lambda+\Delta
u)$, we deduce from (3.36) that $\xi(\xi(2n\lambda+\Delta
u))=(2n\lambda+\Delta u)-2n$. But, it is known [12] that a
$(2n+1)$-dimensional Kenmotsu manifold is locally isometric to the warped
product $I\times_{ce^{t}}N^{2n}$, where $N$ is a Kähler manifold and $I$ is an
open interval. Employing $\xi=\frac{\partial}{\partial t}$ (where $t$ denotes
the coordinate on $I$) into (3.36) we obtain

$\displaystyle\frac{d^{2}(2n\lambda+\Delta u)}{dt}=(2n\lambda+\Delta u)-2n.$

The solution of this can be exhibited as $(2n\lambda+\Delta
u)=A\,cosh\,\,t+B\,sinh\,\,t+2n$, where $A$, $B$ constants on $M$.

Now, suppose that $r\neq-2n(2n+1)$ in some open set $\mathcal{O}$ of $M$, then
from (3.35) we have $Du=\xi(u)\xi$, and this completes the proof. ∎

######  Remark 3.6.

Theorem 1 in [6], Theorem 1 in [5] and Theorem 3 in [7] of Ghosh is a direct
corollary of Theorem 3.2, Theorem 3.3 and Theorem 3.5 respectively.

Now we construct an example of Riemann soliton on 3-diemnsional Kenmotsu
manifold, and verify our results.

######  Example.

Let us indicate the canonical coordinates on $\mathbb{R}^{3}$ by $(x,y,z)$,
and take the 3-dimensional manifold $M\subset\mathbb{R}^{3}$ defined by

$\displaystyle M=\\{(x,y,z)\in\mathbb{R}^{3}|z\neq 0\\}.$

We may easily verify that putting

$\displaystyle\varphi\left(\frac{\partial}{\partial
x}\right)=\frac{\partial}{\partial
y},\quad\varphi\left(\frac{\partial}{\partial
y}\right)=-\frac{\partial}{\partial
x},\quad\varphi\left(\frac{\partial}{\partial z}\right)=0,$
$\displaystyle\xi=\frac{\partial}{\partial z},\quad\eta=dz,\quad
g=\frac{1}{2}exp(2z)(dx^{2}+dy^{2})+dz^{2},$

$(\varphi,\xi,\eta,g)$ is an almost contact metric structure on $M$. The
matrix representation of $g$ with respect to $\frac{\partial}{\partial x}$,
$\frac{\partial}{\partial y}$ and $\frac{\partial}{\partial z}$ is

$\displaystyle(g_{ij})=\begin{pmatrix}\frac{1}{2}exp(2z)&0&0\\\
0&\frac{1}{2}exp(2z)&0\\\ 0&0&1\end{pmatrix}.$

Using the Koszul formula we have

$\displaystyle\begin{gathered}\nabla_{\frac{\partial}{\partial
x}}\frac{\partial}{\partial x}=-\frac{1}{2}exp(2z)\frac{\partial}{\partial
z},\quad\nabla_{\frac{\partial}{\partial x}}\frac{\partial}{\partial
y}=0,\quad\nabla_{\frac{\partial}{\partial x}}\frac{\partial}{\partial
z}=\frac{\partial}{\partial x},\\\ \nabla_{\frac{\partial}{\partial
y}}\frac{\partial}{\partial x}=0,\quad\nabla_{\frac{\partial}{\partial
y}}\frac{\partial}{\partial y}=-\frac{1}{2}exp(2z)\frac{\partial}{\partial
z},\quad\nabla_{\frac{\partial}{\partial y}}\frac{\partial}{\partial
z}=\frac{\partial}{\partial y},\\\ \nabla_{\frac{\partial}{\partial
z}}\frac{\partial}{\partial x}=\frac{\partial}{\partial
x},\quad\nabla_{\frac{\partial}{\partial z}}\frac{\partial}{\partial
y}=\frac{\partial}{\partial y},\quad\nabla_{\frac{\partial}{\partial
x}}\frac{\partial}{\partial z}=0,\end{gathered}$ (3.40)

where $\nabla$ denotes the Levi-Civita connection of the Riemannian metric
$g$. It is not hard to verify that the condition (3.1) for Kenmotsu manifold
is satisfied. Hence, the manifold under consideration is a Kenmotsu manifold.
By a straightforward calculation we have

$\displaystyle R\left(\frac{\partial}{\partial x},\frac{\partial}{\partial
y}\right)\frac{\partial}{\partial z}=0,\quad R\left(\frac{\partial}{\partial
x},\frac{\partial}{\partial z}\right)\frac{\partial}{\partial y}=0,\quad
R\left(\frac{\partial}{\partial x},\frac{\partial}{\partial
z}\right)\frac{\partial}{\partial z}=-\frac{\partial}{\partial x},$
$\displaystyle R\left(\frac{\partial}{\partial x},\frac{\partial}{\partial
y}\right)\frac{\partial}{\partial
x}=\frac{1}{2}exp(2z)\frac{\partial}{\partial y},\quad
R\left(\frac{\partial}{\partial x},\frac{\partial}{\partial
y}\right)\frac{\partial}{\partial
y}=-\frac{1}{2}exp(2z)\frac{\partial}{\partial x},$ $\displaystyle
R\left(\frac{\partial}{\partial x},\frac{\partial}{\partial
z}\right)\frac{\partial}{\partial
x}=\frac{1}{2}exp(2z)\frac{\partial}{\partial z},\quad
R\left(\frac{\partial}{\partial y},\frac{\partial}{\partial
z}\right)\frac{\partial}{\partial x}=0,$ $\displaystyle
R\left(\frac{\partial}{\partial y},\frac{\partial}{\partial
z}\right)\frac{\partial}{\partial
y}=\frac{1}{2}exp(2z)\frac{\partial}{\partial z},\quad
R\left(\frac{\partial}{\partial y},\frac{\partial}{\partial
z}\right)\frac{\partial}{\partial z}=-\frac{\partial}{\partial y},$ (3.41)

and applying these, we obtain

$\displaystyle S\left(\frac{\partial}{\partial x},\frac{\partial}{\partial
x}\right)=-exp(2z),\quad S\left(\frac{\partial}{\partial
y},\frac{\partial}{\partial y}\right)=-exp(2z),\quad
S\left(\frac{\partial}{\partial z},\frac{\partial}{\partial z}\right)=-2.$

Thus, the Ricci tensor satisfy $S(X,Y)=-2ng(X,Y)$ for any $X,Y\in\Gamma(TM)$,
that is, $M$ is Einstein. From (3.41), we can easily shows that

$\displaystyle R(X,Y)Z=-\\{g(Y,Z)X-g(X,Z)Y\\},$ (3.42)

for any $X,Y,Z\in\Gamma(TM)$. Next consider a vector field

$\displaystyle V=a\left(y\frac{\partial}{\partial x}-x\frac{\partial}{\partial
y}\right),$ (3.43)

where $a\neq 0$ is a constant. One can easily verify that $V$ has constant
divergence. As a result of (3.40), one can easily justify that

$\displaystyle(\pounds_{V}g)(X,Y)=0,$ (3.44)

for any $X,Y\in\Gamma(TM)$. Unifying (3.44) and (3.42), we obtain that $g$ is
a Riemann soliton, that is, (1.2) holds true with $V$ as in (3.43) and
$\lambda=1$. Further (3.42) shows that $M$ is of constant negative curvature
$-1$ and this verifies the Theorem 3.3.

##  4\. Riemann solitons and almost gradient Riemann solitons on
$(\kappa,\mu)^{\prime}$-almost Kenmotsu manifolds with $\kappa<-1$

If the Reeb or characterstic vector field $\xi$ of an almost Kenmotsu manifold
$M$ belonging to the $(\kappa,\mu)^{\prime}$-nullity distribution (see [3]),
that is,

$\displaystyle
R(X,Y)\xi=\kappa\\{\eta(Y)X-\eta(X)Y\\}+\mu\\{\eta(Y)h^{\prime}X-\eta(X)h^{\prime}Y\\},$
(4.1)

for some constants $\kappa$ and $\mu$, then $M$ is called
$(\kappa,\mu)^{\prime}$-almost Kenmotsu manifold. Classification of almost
Kenmotsu manifold with $\xi$ belonging to the $(\kappa,\mu)^{\prime}$-nulliy
distribution have done by many geometers. For more detail, we refer to [3, 13,
19, 20]. On $(\kappa,\mu)^{\prime}$-almost Kenmotsu manifold, the following
relation hold

$\displaystyle h^{\prime 2}=(\kappa+1)\varphi^{2},$
$\displaystyle\quad\text{or}\quad h^{2}=(\kappa+1)\varphi^{2},$ (4.2)
$\displaystyle Q\xi=2n\kappa\xi.$ (4.3)

Let $X\in Ker\eta$ be an eigenvector field of $h^{\prime}$ orthogonal to $\xi$
with the corresponding eigenvalue $\theta$. As a result of (4.2), we get
$\theta^{2}=-(\kappa+1)$ and hence we have $\kappa\leq-1$. From (4.2) we
remark that the tensor field $h^{\prime}=0$ (equivalently, $h=0$) if and only
if $\kappa=-1$, and $\kappa<-1$ if and only if $h^{\prime}\neq 0$. As stated
by Dileo and Pastore [3] in Proposition 4.1 that, on a
$(\kappa,\mu)^{\prime}$-almost Kenmotsu manifold with $\kappa<-1$, we have
$\mu=-2$. In this section, we paln to investigate the geometry of a Riemann
soliton and gradient almost Riemann soliton on non-Kenmotsu
$(\kappa,\mu)^{\prime}$-almost Kenmotsu manifold. First, we reminisce the
following result for our later use.

######  Lemma 4.1.

In a $(\kappa,\mu)^{\prime}$-almost Kenmotsu manifold $M$ with $\kappa<-1$,
the Ricci operator satisfies

$\displaystyle QX=-2nX+2n(\kappa+1)\eta(X)\xi-2nh^{\prime},$ (4.4)

where $h^{\prime}\neq 0$. Moreover, the scalar curvature $r$ of $M$ is
$2n(\kappa-2n)$.

Now, we prove the following outcome.

######  Theorem 4.2.

Let $M$ be a non-Kenmotsu $(\kappa,\mu)^{\prime}$-almost Kenmotsu manifold. If
metric of $M$ is a Riemann soliton with $divV=constant$, then either the
manifold is locally isometric to the Reimannian product
$\mathbb{H}^{n+1}(-4)\times\mathbb{R}^{n}$ or the potential vector field $V$
is strict infinitesimal contact transformation.

###### Proof.

Contracting the equation (1.2) over $X$ and $W$, one can get

$\displaystyle(\pounds_{V}g)(Y,Z)+\frac{2}{2n-1}S(Y,Z)+\frac{2}{2n-1}(2n\lambda+divV)g(Y,Z)=0.$

As a result of Lemma 4.1, the above equation transform into

$\displaystyle(\pounds_{V}g)(Y,$ $\displaystyle
Z)+\frac{2}{2n-1}\\{(2n\lambda+divV-2n)g(Y,Z)$
$\displaystyle+2n(\kappa+1)\eta(Y)\eta(Z)-2ng(h^{\prime}Y,Z)\\}=0.$ (4.5)

Differentiating (4.5) covariantly along $X$, recalling (2.3) and $V$ has a
constant divergence, we have

$\displaystyle(\nabla_{X}\pounds_{V}g)(Y,Z)=-$
$\displaystyle\frac{4n}{2n-1}(\kappa+1)\\{g(X+h^{\prime}X,Y)\eta(Z)+g(X+h^{\prime}X,Z)\eta(Y)$
$\displaystyle-2\eta(X)\eta(Y)\eta(Z)\\}+\frac{4n}{2n-1}g((\nabla_{X}h^{\prime})Y,Z).$
(4.6)

In [3] Dileo and Pastore proved that on $M$ there holds

$\displaystyle g((\nabla_{X}h^{\prime})Y,Z)=g$
$\displaystyle((\kappa+1)X-h^{\prime}X,Y)\eta(Z)+\eta(Y)g((\kappa+1)X-h^{\prime}X,Z)$
$\displaystyle-2(\kappa+1)\eta(X)\eta(Y)\eta(Z).$ (4.7)

Making use of (4.6) into (2.6), we entails that

$\displaystyle(\pounds_{V}\nabla)(X,Y)=-\frac{4n}{2n-1}(\kappa+2)g(h^{\prime}X,Y)\xi,$
(4.8)

where we applied (4.7). Taking covariant differentiation of (4.8) we get

$\displaystyle(\nabla_{Y}\pounds_{V}\nabla)(X,Z)=$
$\displaystyle-\frac{4n}{2n-1}(\kappa+2)g((\nabla_{Y}h^{\prime})Y,Z)\xi$
$\displaystyle-\frac{4n}{2n-1}(\kappa+2)g(h^{\prime}X,Z)(Y-\eta(Y)\xi+h^{\prime}Y).$

Putting the foregoing equation into (2.7) yields

$\displaystyle(\pounds_{V}$ $\displaystyle
R)(X,Y)Z=\frac{4n}{2n-1}\\{g((\nabla_{Y}h^{\prime})X,Z)\xi-g((\nabla_{X}h^{\prime})Y,Z)\xi$
$\displaystyle+g(h^{\prime}X,Z)(Y-\eta(Y)\xi+h^{\prime}Y)-g(h^{\prime}Y,Z)(X-\eta(X)\xi+h^{\prime}X)\\}.$
(4.9)

Contracting the equation (4.9) with respect to $X$ and recalling (4.7) we
easily obtain

$\displaystyle(\pounds_{V}S)(Y,Z)=-\frac{8n^{2}}{2n-1}(\kappa+2)g(h^{\prime}Y,Z).$
(4.10)

From (4.4), the Ricci tensor can be written as

$\displaystyle
S(Y,Z)=-2ng(Y,Z)+2n(\kappa+1)\eta(Y)\eta(Z)-2ng(h^{\prime}Y,Z).$

Lie-derivative of this equation along potential vector field $V$, utilization
of (2.3) and (4.5) yields

$\displaystyle(\pounds_{V}S)(Y,Z)$
$\displaystyle=\frac{4n}{2n-1}\\{(2n\lambda+divV+2n\kappa)g(Y,Z)-2(\kappa+1)(2n\lambda+divV$
$\displaystyle+2n\kappa)\eta(Y)\eta(Z)+(2n\lambda+divV-4n)g(h^{\prime}Y,Z)\\}+2n(\kappa+1)$
$\displaystyle\\{\eta(Y)g(\pounds_{V}\xi,Z)+\eta(Z)g(\pounds_{V}\xi,Y)\\}-2ng((\pounds_{V}h^{\prime})Y,Z).$
(4.11)

Putting (4.11) together with (4.10), we get

$\displaystyle g((\pounds_{V}h^{\prime})Y,Z)=$
$\displaystyle\frac{2}{2n-1}\\{(2n\lambda+divV+2n\kappa)g(Y,Z)-2(\kappa+1)(2n\lambda+divV$
$\displaystyle+2n\kappa)\eta(Y)\eta(Z)+(2n(\kappa+2)+2n\lambda+divV-4n)g(h^{\prime}Y,Z)\\}$
$\displaystyle+(\kappa+1)\\{\eta(Y)g(\pounds_{V}\xi,Z)+\eta(Z)g(\pounds_{V}\xi,Y)\\}.$
(4.12)

Note that by switching $Y=Z=\xi$ in (4.5) we obtain
$\eta(\pounds_{V}\xi)-\frac{1}{2n-1}(2n\lambda+divV+2n\kappa)=0$. Applying
this equation and substituting $Y=Z=\xi$ in (4.12) one can get
$(2n\lambda+divV+2n\kappa)=0$ by reason of $\kappa<-1$ and eventually we have

$\displaystyle(\pounds_{V}h^{\prime})Y=(\kappa+1)\\{\eta(Y)\pounds_{V}\xi+g(\pounds_{V}\xi,Y)\xi\\}.$

Inserting $Y$ by $\xi$ in the foregoing equation and utilization of
$(2n\lambda+\Delta u+2n\kappa)=0$ gives

$\displaystyle h^{\prime}\pounds_{V}\xi=-(\kappa+1)\pounds_{V}\xi.$ (4.13)

In view of (4.2), $(2n\lambda+divV+2n\kappa)=0$ and $\kappa<-1$, the action of
$h^{\prime}$ on the above equation gives
$h^{\prime}\pounds_{V}\xi=\pounds_{V}\xi$. This together with (4.13) yields

$\displaystyle(\kappa+2)\pounds_{V}\xi=0.$

Suppose $\kappa=-2$, then it follows from Proposition 4.1 and Corollary 4.2 of
Dielo and Pastore [3] that a non-Kenmotsu $(\kappa,\mu)^{\prime}$-almost
Kenmotsu manifold is locally isometric to the Riemannian product
$\mathbb{H}^{n+1}(-4)\times\mathbb{R}^{n}$. If $\kappa\neq-2$, then we have
$\pounds_{V}\xi=0$. As a result of $2n\lambda+divV+2n\kappa=0$ and
$(\pounds_{V}g)(X,\xi)=0$ we obtain

$\displaystyle(\pounds_{V}\eta)X=(\pounds_{V}g)(X,\xi)+g(X,\pounds_{V}\xi)=0.$

This means potential vector field $V$ is strict infinitesimal contact
transformation. This finishes the proof. ∎

As a result of Lemma 4.1, the relation (3.27) and (3.26), we obtain the
following fruitful result.

######  Theorem 4.3.

Let $M$ be a non-Kenmotsu $(\kappa,\mu)^{\prime}$-almost Kenmotsu manifold
which admits a gradient almost Riemann soliton. Then, the soliton is expanding
with $\lambda=\frac{6n-2}{2n-1}$ and $M$ is locally isometric to the
Riemannian product $\mathbb{H}^{n+1}(-4)\times\mathbb{R}^{n}$. Moreover, the
potential vector field is tangential to the Euclidean factor $\mathbb{R}^{n}$.

###### Proof.

With the help of (4.4) we have

$\displaystyle(\nabla_{Y}Q)X-(\nabla_{X}Q)Y=$
$\displaystyle-2n((\nabla_{Y}h^{\prime})X-(\nabla_{X}h^{\prime})Y)$
$\displaystyle-2n(\kappa+1)(\eta(Y)(X+h^{\prime}X)-\eta(X)(Y+h^{\prime}Y)).$

Utilization of the above equation in (3.26) gives

$\displaystyle R(X,Y)Du=$
$\displaystyle-\frac{2n}{2n-1}\\{((\nabla_{Y}h^{\prime})X-(\nabla_{X}h^{\prime})Y)$
$\displaystyle+(\kappa+1)(\eta(Y)(X+h^{\prime}X)-\eta(X)(Y+h^{\prime}Y))\\}$
$\displaystyle+\frac{1}{2n-1}\\{Y(2n\lambda+\Delta u)X-X(2n\lambda+\Delta
u)Y\\}$ (4.14)

Using $\xi$ in place of $X$ in (4.14) we get an equation, then taking inner
product of the resulting equation with $\xi$ gives

$\displaystyle g(R(\xi,Y)Du,\xi)=\frac{1}{2n-1}\\{Y(2n\lambda+\Delta
u)-\xi(2n\lambda+\Delta u)\eta(Y)\\}.$ (4.15)

As a result of (4.1), we get

$\displaystyle
g(R(\xi,Y)\xi,Du)=\kappa\\{\xi(u)\eta(Y)-Y(u)\\}+2g(h^{\prime}Du,Y),$ (4.16)

where we used $\mu=-2$. Comparing (4.15) with (4.16) yields that

$\displaystyle\frac{1}{2n-1}D(2n\lambda+\Delta u)=\kappa
Du-\kappa\xi(u)\xi-2h^{\prime}Du+\frac{1}{2n-1}\xi(2n\lambda+\Delta u)\xi.$
(4.17)

According to Corollary 4.1 of Delio and Pastore [3], and Lemma 3.4 of Wang and
Liu [21], we have that $tr(\nabla_{X}h^{\prime})=0$ and
$(divh^{\prime})X=2n(\kappa+1)\eta(X)$. Applying these equation and
contracting (4.14) over $X$ gives that

$\displaystyle S(Y,Du)=\frac{2n}{2n-1}Y(2n\lambda+\Delta u).$

As a result of (4.4), the above equation transform into

$\displaystyle\frac{1}{2n-1}D(2n\lambda+\Delta
u)=-Du+(\kappa+1)\xi(u)\xi-h^{\prime}Du.$ (4.18)

By virtue of (4.17) and (4.18) we obtain

$\displaystyle(\kappa+1)Du-(2\kappa+1)\xi(u)\xi-h^{\prime}Du+\frac{1}{2n-1}\xi(2n\lambda+\Delta
u)=0.$

Remembering the assumption $\kappa<-1$ and equation (4.2), the action of
$h^{\prime}$ on above equation gives that

$\displaystyle h^{\prime}Du=-Du+\xi(u)\xi.$ (4.19)

Again making use of (4.2), the action $h^{\prime}$ on (4.19) yields
$(\kappa+1)(Du-\xi(u)\xi)=h^{\prime}Du$. This together with (4.19) gives that

$\displaystyle(\kappa+2)(Du-\xi(u)\xi)=0.$ (4.20)

Thus, we have either $\kappa=-2$ or $Du=\xi(u)\xi$. Now, we prove that the
second case can not occur. In fact, if we assume that the second case is true,
that is, $Du=\xi(u)\xi$, then covariant derivative of this along $X$ gives

$\displaystyle\nabla_{X}Du=X(\xi(u))\xi+\xi(u)\\{X-\eta(X)\xi+h^{\prime}X\\}.$

With the aid of above equation, (3.27) becomes

$\displaystyle\frac{1}{2n-1}QX=-(\xi(u)+\frac{1}{2n-1}(2n\lambda+\Delta
u))X+(\xi(u)\eta(X)-X(\xi(u)))\xi-\xi(u)h^{\prime}X.$

Utilization of (4.4) in the foregoing equation furnishes

$\displaystyle\left(\xi(u)+\frac{1}{2n-1}(2n\lambda+\Delta
u-2n)\right)X+\left(\xi(u)-\frac{2n}{2n-1}\right)h^{\prime}X$
$\displaystyle+\left(\frac{2n}{2n-1}(\kappa+1)\eta(X)+X(\xi(u))-\xi(u)\eta(X)\right)\xi=0.$
(4.21)

Contracting (4.21) with respect to $X$ and recalling (2.4) we obtain

$\displaystyle 2n(\kappa+1)\left(\xi(u)-\frac{2n}{2n-1}\right)=0,$

and this exhibits that $\xi(u)=\frac{2n}{2n-1}$ because of $\kappa<-1$.
Putting $\xi(u)=\frac{2n}{2n-1}$ into (4.21) we obatin

$\displaystyle(2n\lambda+\Delta u)X+2n\kappa\eta(X)\xi=0.$ (4.22)

Considering $X$ in (4.22) is orthogonal to $\xi$, we have $2n\lambda+\Delta
u=0$. Thus, (4.22) becomes $2n\kappa\eta(X)\xi=0$. It follows that $\kappa=0$,
this contradicts our assumption $\kappa<-1$. Therefore, we obtain from (4.20)
that $\kappa=-2$. In view of $\kappa=-2=\mu$, on the basis of results of Dileo
and Pastore [3] we obtain that $M$ is locally isometric to the Riemannian
product $\mathbb{H}^{n+1}(-4)\times\mathbb{R}^{n}$. Employing $\kappa=-2$ and
(4.19) in (4.17) we obtain

$\displaystyle D(2n\lambda+\Delta u)=\xi(2n\lambda+\Delta u).$ (4.23)

Utilizing (4.23) and (4.19) in (4.18) provides

$\displaystyle\xi\left(\frac{1}{2n-1}(2n\lambda+\Delta u)+2u\right)=0.$ (4.24)

In the framework of (4.4), (3.27) and (4.24) we acquire that

$\displaystyle 2n\lambda+\Delta u=4n+\frac{1}{2}\xi(\xi(2n\lambda+\Delta u)).$
(4.25)

From (4.2) we have $h^{\prime 2}=-\varphi^{2}$. Now, we shall indicate by
$[1]^{\prime}$ and $[-1]^{\prime}$ the distribution of the eigen vectors of
$h^{\prime}$ orthogonal to $\xi$ with eigen values $1$ and $-1$ respectively,
and also we may consider a local orthonormal $\varphi$-frame $\\{E_{i},\varphi
E_{i},\xi\\}_{i=1}^{n}$ with $E_{i}\in[1]^{\prime}$ and $\varphi
E_{i}\in[-1]^{\prime}$. As a result of (4.19), we easily find that $Du$ has no
components on the distribution $[1]^{\prime}$. Thus, we write
$Du=\sum_{i=n}^{n}\gamma_{i}\varphi E_{i}+\xi(u)\xi$, where
$\\{\gamma_{i}\\}_{i=1}^{n}$ are smooth function on $M$. Applying this and
(4.24) in (3.27) we obtain that

$\displaystyle\frac{1}{2n-1}QX=$
$\displaystyle\frac{1}{2n-1}\left(\frac{1}{2}\xi(2n\lambda+\Delta
u)-(2n\lambda+\Delta u)\right)X-\sum_{i=1}^{n}X(\gamma_{i})\varphi E_{i}$
$\displaystyle-\sum_{i=1}^{n}\gamma_{i}\nabla_{X}\varphi
E_{i}+\frac{1}{2(2n-1)}\xi(2n\lambda+\Delta u)h^{\prime}X$
$\displaystyle+\frac{1}{2(2n-1)}\left(X(\xi(2n\lambda+\Delta
u))-\xi(2n\lambda+\Delta u)\eta(X)\right)\xi.$

By virtue of (4.4), the above equation transform into

$\displaystyle\frac{1}{2n-1}$
$\displaystyle\left(\frac{1}{2}\xi(2n\lambda+\Delta u)-(2n\lambda+\Delta
u)+2n\right)X-\sum_{i=1}^{n}X(\gamma_{i})\varphi
E_{i}-\sum_{i=1}^{n}\gamma_{i}\nabla_{X}\varphi E_{i}$
$\displaystyle+\frac{1}{2n-1}\left(\frac{1}{2}\xi(2n\lambda+\Delta
u)+2n\right)h^{\prime}X+\frac{1}{2(2n-1)}\\{X(\xi(2n\lambda+\Delta u))$
$\displaystyle-\xi(2n\lambda+\Delta u)\eta(X)+4n\eta(X)\\}\xi=0.$ (4.26)

According to the proof of Proposition 4.1 of [3], we have that
$\nabla_{E_{j}}\varphi E_{i}\in[-1]^{\prime}$ for any $E_{i}\in[1]^{\prime}$,
$1\leq j\leq n$. Consequently, using $E_{j}\in[1]^{\prime}$ in place of $X$ in
(4.26) provides

$\displaystyle(2n\lambda+\Delta u)=\xi(2n\lambda+\Delta u)+4n.$ (4.27)

Unifying (4.25) with (4.27) yields that $2n\lambda+\Delta u=4n$. Now,
contracting (3.27) over $X$ gives

$\displaystyle\Delta u=-\frac{4n^{2}}{2n-1},$

where we applied $\kappa=-2$ and $r=2n(\kappa-2n)$. Combining the above
equation with $2n\lambda+\Delta u=4n$ yields that
$\lambda=\frac{6n-2}{2n-1}>0$, and this means that the gradient almost Riemann
soliton is expanding. Eventually, utilization of $2n\lambda+\Delta u=4n$ in
(4.24) gives $\xi(u)=0$ and thus from (4.19) we have $h^{\prime}Du=-Du$.
According to Theorem 4.2 of [3], the factor $\mathbb{R}^{n}$ in the product
space $\mathbb{H}^{n+1}(-4)\times\mathbb{R}^{n}$ is the integral submanifold
of the distribution $[-1]^{\prime}$. Therefore, $h^{\prime}Du=-Du$ implies
that the gradient of the potential function $Du$ is tangential to the
Euclidean factor $\mathbb{R}^{n}$. ∎

######  Remark 4.4.

Theorem 3.1 of Wang [20] is a direct corollary of the above Theorem 4.3.

Before closing this section, we present an example of 3-dimensional
$(\kappa,\mu)^{\prime}$-almost Kenmotsu manifolds admitting a Riemann soliton.

######  Example.

We consider a 3-dimensional manifold

$\displaystyle M=\\{(x,y,z)\in\mathbb{R}^{3},z\neq 0\\},$

and linear independently vector fields

$\displaystyle e_{1}=-\frac{\partial}{\partial x}+2y\frac{\partial}{\partial
y}-\frac{\partial}{\partial z},\quad e_{2}=\frac{\partial}{\partial y},\quad
e_{3}=\frac{\partial}{\partial z}.$

One can easily check that

$\displaystyle[e_{1},e_{2}]=-2e_{2},\quad[e_{2},e_{3}]=0,\quad[e_{1},e_{3}]=0.$

On $M$ we define a (1,1)-tensor field $\varphi$ by $\varphi(e_{1})=0$,
$\varphi(e_{2})=e_{3}$ and $\varphi(e_{3})=-e_{2}$, and we define a Riemannian
metric $g$ such that $g(e_{i},e_{j})=\delta_{ij}$, $1\leq i,j\leq 3$. We
denote by $\xi=e_{1}$ and $\eta$ its dual 1-form with respect to the metric
$g$.

On the other hand, it is not hard to justify that $M$ with the structure
$(\varphi,\xi,\eta,g)$ is an almost Kenmotsu structure, which is not Kenmotsu,
since the operator $h^{\prime}$ does not vanish. In fact, it is given by

$\displaystyle h^{\prime}(e_{1})=0,\quad h^{\prime}(e_{2})=e_{2},\quad
h^{\prime}(e_{3})=-e_{3}.$

Utilization of Koszul formula gives

$\displaystyle\begin{gathered}\nabla_{e_{1}}e_{1}=0,\quad\nabla_{e_{1}}e_{2}=0,\quad\nabla_{e_{1}}e_{3}=0,\\\
\nabla_{e_{2}}e_{1}=2e_{2},\quad\nabla_{e_{2}}e_{2}=-2e_{1},\quad\nabla_{e_{2}}e_{3}=0,\\\
\nabla_{e_{3}}e_{1}=0,\quad\nabla_{e_{3}}e_{2}=0,\quad\nabla_{e_{3}}e_{3}=0,\end{gathered}$
(4.31)

where $\nabla$ denotes the Levi-Civita connection of the Riemannian metric
$g$. By a straightforward calculation we have

$\displaystyle\begin{gathered}R(e_{1},e_{2})e_{1}=4e_{2},\quad
R(e_{1},e_{2})e_{2}=-4e_{1},\quad R(e_{1},e_{2})e_{3}=0,\\\
R(e_{1},e_{3})e_{1}=0,\quad R(e_{1},e_{3})e_{2}=0,\quad
R(e_{1},e_{3})e_{3}=0,\\\ R(e_{2},e_{3})e_{1}=0,\quad
R(e_{2},e_{3})e_{2}=0,\quad R(e_{2},e_{3})e_{3}=0.\end{gathered}$ (4.35)

With the help of the expressions of the curvature tensor, we conclude that the
characteristic vector field $\xi$ belongs to the
$(\kappa,\mu)^{\prime}$-nullity distribution with $\kappa=-2$ and $\mu=-2$.

Let us consider a vector field

$\displaystyle V=e^{-2x}\frac{\partial}{\partial
y}+4(x-z)\frac{\partial}{\partial z}.$ (4.36)

One can easily check that $divV=-4$, a constant. As a result of (4.31), one
can get

$\displaystyle\begin{gathered}(\pounds_{V}g)(e_{1},e_{1})=0,\quad(\pounds_{V}g)(e_{2},e_{2})=0,\quad(\pounds_{V}g)(e_{3},e_{3})=-8,\\\
(\pounds_{V}g)(e_{1},e_{2})=0,\quad(\pounds_{V}g)(e_{1},e_{3})=0,\quad(\pounds_{V}g)(e_{2},e_{3})=0\end{gathered}$
(4.39)

In view of (4.39) and (4.35), one can easily verify that

$\displaystyle
2R(e_{i},e_{j},e_{k},e_{l})+4(g\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}g)(e_{i},e_{j},e_{k},e_{l})+(g\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}\pounds_{V}g)(e_{i},e_{j},e_{k},e_{l})=0,$

for $1\leq i,j,k,l\leq 3$. Thus $g$ is a Riemann soliton with the soliton
vector field $V$ as given in (4.36) and $\lambda=4$. According to Dileo and
Pastore [3], we obtain that $(-2,-2)^{\prime}$-almost Kenmotsu manifold $M$ is
locally isometric to the product $\mathbb{H}^{2}(-4)\times\mathbb{R}$. This
verifies Theorem 4.2.

## References

  * [1] D.E. Blair, Riemannian geometry of contact and symplectic manifolds. In: Progress in Mathematics, 203. Birkhäuser, Boston (2010).
  * [2] G. Dileo, A. M. Pastore, Almost Kenmotsu manifolds and local symmetry, Bull. Belg. Math. Soc. Simon Stevin. 14(2) (2007), 343-354.
  * [3] G. Dileo, A. M. Pastore, Almost Kenmotsu manifolds and nullity distributions, J. Geom. 93(1-2) (2009), 46-61.
  * [4] M.N. Devaraja, H.A. Kumara, V. Venkatesha,  Riemann soliton within the framework of contact geometry, Quaestiones Mathematicae, (2020) DOI: 10.2989/16073606.2020.1732495
  * [5] A. Ghosh, Kenmotsu 3-metric as a Ricci soliton, Chaos Solitons & Fractals, 44 (2011), 647-650.
  * [6] A. Ghosh, An $\eta$-Einstein Kenmotsu metric as a Ricci soliton, Publ. Math. Debrecen, 82 (2013), 691-598.
  * [7] A. Ghosh, Ricci soliton and Ricci almost soliton within the framework of Kenmotsu manifold, Carpathian Math. Publ. 11(1) (2019), 59-69.
  * [8] D. Janssens, L. Vanhecke, Almost contact structures and curvature tensors, Kodai Math. J. 4(1) (1981), 1-27.
  * [9] R.S, Hamilton, The Ricci flow on surfaces. Mathematics and general relativity. Contemp. Math., Amer. Math. Soc. 71 (1988), 237-262.
  * [10] I.E. Hiric̆a, C. Udrişste, Ricci and Riemann solitons, Balkan J. Geom. Applications. 21(2) (2016), 35-44.
  * [11] I.E. Hiric̆a, C. Udrişste, Basic evolution PDE’s in Riemannian Geometry, Balkan J.Geom.Appl. 17(1) (2012), 30-40.
  * [12] K. Kenmotsu, A class of almost contact Riemannian manifolds, Tôhoku Math. J. 24(1) (1972), 93-103.
  * [13] D.G. Prakasha, P. Veeresha, Venkatesha, The Fischer–Marsden conjecture on non-Kenmotsu $(\kappa,\mu)^{\prime}$-almost Kenmotsu manifolds, J. Geom. 110 (1) (2019), https://doi.org/10.1007/s00022-018-0457-8.
  * [14] S.E. Stepanov, I.I. Tsyganok, The theory of infinitesimal harmonic transformations and its applications to the global geometry of Riemann solitons, Balk. J. Geom. Appl. 24 (2019), 113–121.
  * [15] Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc. 117 (1965), 251-275.
  * [16] C. Udrişte, Riemann flow and Riemann wave, Ann. Univ. Vest, Timisoara. Ser. Mat.-Inf. 48(1-2) (2010), 265-274.
  * [17] C. Udrişte, Riemann flow and Riemann wave via bialternate product Riemannian metric. preprint, arXiv.org/math.DG/1112.4279v4 (2012).
  * [18] Venkatesha, D. M. Naik, H. A. Kumara, $*$-Ricci soliton and gradient almost $*$-Ricci soliton on Kenmotsu manifolds, Mathemtica Slovoca (accepted).
  * [19] V. Venkatesha, H. A. Kumara, Gradient $\rho$-Einstein soliton on almost Kenmotsu manifolds, Ann. Univ. Ferrara. 65(2) (2019), 375-388.
  * [20] Y. Wang, Gradient almost Ricci soliton on two classes of almost Kenmotsu manifolds, J. Korean Math. Soc. 53(5) (2016), 1101-1114.
  * [21] Y. Wang, X. Liu, Locally symmetric CR-integrable almost Kenmotsu manifolds, Mediterr. J. Math. 12(1) (2015), 159-171.
  * [22] K. Yano, Integral formulas in Riemannian geometry, Marcel Dekker, New York, 1970.


# New versions of the Wallis-Fon-Der-Flaass construction to create divisible
design graphs

Vladislav V. Kabanov<EMAIL_ADDRESS>Krasovskii Institute of Mathematics and
Mechanics, S. Kovalevskaja st. 16, Yekaterinburg, 620990, Russia

###### Abstract

A $k$-regular graph on $v$ vertices is a divisible design graph with
parameters $(v,k,\lambda_{1},\lambda_{2},m,n)$ if its vertex set can be
partitioned into $m$ classes of size $n$, such that any two different vertices
from the same class have $\lambda_{1}$ common neighbours, and any two vertices
from different classes have $\lambda_{2}$ common neighbours whenever it is not
complete or edgeless. If $m=1$, then a divisible design graph is strongly
regular with parameters $(v,k,\lambda_{1},\lambda_{1})$. In this paper the
Wallis-Fon-Der-Flaass construction of strongly regular graphs is modified to
create new constructions of divisible design graphs. In some cases, these
constructions lead to strongly regular graphs.

###### keywords:

Divisible design graph, Strongly regular graph

###### MSC:

[2020] 05B20, 05C50, 05E30

††journal: Discrete Mathematics

##  1 Introduction

A $k$-regular graph on $v$ vertices is a divisible design graph with
parameters $(v,k,\lambda_{1},\lambda_{2},m,n)$ if its vertex set can be
partitioned into $m$ classes of size $n$, such that any two different vertices
from the same class have $\lambda_{1}$ common neighbours, and any two vertices
from different classes have $\lambda_{2}$ common neighbours whenever it is not
complete or edgeless. The partition of a divisible design graph into classes
is called a canonical partition. If $m=1$, then a divisible design graph is a
strongly regular with parameters $(v,k,\lambda_{1},\lambda_{1})$. In this
paper the Wallis-Fon-Der-Flaass construction of strongly regular graphs is
modified to create new constructions of divisible design graphs. In some
cases, these constructions lead to strongly regular graphs.

There is a connection between divisible design graphs and divisible designs
since their vertices can be considered as points, and the neighbourhoods of
the vertices as blocks. Obviously, such a divisible design is symmetric, since
it has a symmetric incidence matrix with zero diagonal. R.C. Bose and W.S.
Connor studied the combinatorial properties of divisible designs in [2].

Divisible design graphs were first introduced by W.H. Haemers, H. Kharaghani
and M. Meulenberg in [10]. In particular, the authors have proposed twenty
constructions of divisible design graphs using various combinatorial
structures. Some new divisible design graphs were constructed by D. Crnković
and W.H. Haemers in [6].

The spectrum of any divisible design graph with parameters
$(v,k,\lambda_{1},\lambda_{2},m,n)$ can be calculated using its parameters as
follows:

$\\{k^{1},\sqrt{k-\lambda_{1}}^{f_{1}},-\sqrt{k-\lambda_{1}}^{f_{2}},\sqrt{k^{2}-\lambda_{2}v}^{g_{1}},-\sqrt{k^{2}-\lambda_{2}v}^{g_{2}}\\},$

where exponents are eigenvalue multiplicities. Moreover, $f_{1}+f_{2}=m(n-1)$
and $g_{1}+g_{2}=m-1$ [10, Lemma 2.1]. The eigenvalues for all divisible
design graphs in this article are calculated using this result.

W.D. Wallis proposed in [18] a new construction of strongly regular graphs
based on an affine designs and a Steiner 2-design. Later D.G. Fon-Der-Flaass
found how to modify a partial case of Wallis construction, when the
corresponding Steiner $2$-design has blocks of size $2$, in order to obtain
hyperexponentially many strongly regular graphs with the same parameters [7].
M. Muzychuk in [15] showed how to modify Fon-Der-Flaass ideas in order to
cover all the cases of Wallis construction. Moreover, he showed that a Steiner
2-design in the original Wallis construction can be replaced by a partial
linear space and discovered new prolific constructions of strongly regular
graphs. Some of the ideas of the above constructions turned out to be useful
for new constructions of divisible design graphs as well. Divisible design
graphs are constructed in this paper by modification of the Wallis-Fon-Der-
Flaass construction of strongly regular graphs.

Just like W.D. Wallis and D.G. Fon-Der-Flaass we use affine designs for our
construction.

An affine design $\mathcal{D}=(\mathcal{P},\mathcal{B})$ with parameters $q$
and $r$ is a design, where $\mathcal{P}$ is a set of points and $\mathcal{B}$
is a set of blocks, with the following two properties:

  * $(i)$

every two blocks are either disjoint or intersect in $r$ points;

  * $(ii)$

each block together with all blocks disjoint from it forms a parallel class: a
set of $q$ mutually disjoint blocks partitioning all points of the design.

If $\mathcal{D}=(\mathcal{P},\mathcal{B})$ is an affine design with parameters
$q$ and $r$, then the number $\varepsilon:=(r-1)/(q-1)$ is integer and all
parameters of $D$ are expressed in terms of $q$ and $r$ [7, Lemma 1]:

$v$ | $q^{2}r$ | the number of points;
---|---|---
$b$ | $q^{3}\varepsilon+q^{2}+q$ | the number of blocks;
$m$ | $q^{2}\varepsilon+q+1$ | the number of parallel classes;
$k$ | $qr$ | the block size;
$\lambda$ | $q\varepsilon+1$ | the number of blocks containing
|  | any 2 distinct points.

Any $d$-dimensional affine space over a finite field of order $q$ with all
hyperplanes in the form of blocks is an affine design with $r=q^{d-2}$. This
design has $q^{d}$ points, any block contains $q^{d-1}$ points, $q^{d-2}$
points are on the intersection of any two different blocks, and the number of
blocks containing any 2 distinct points is $(q^{d-1}-1)/(q-1)$. Other known
examples are Hadamard $3$-designs, where $q=2$.

A Latin square of side $m$ is an $m\times m$ array in which each cell contains
a single symbol from an $m$-set, such that each symbol occurs exactly once in
each row and exactly once in each column. A Latin square
$\mathcal{L}=(e(i,j))_{m\times m}$ of side $m$ is symmetric if $e(i,j)=e(j,i)$
for all $i,j\in[m]$. A Latin square of side $m$ on the integers set
$[m]:=\\{1,2,\dots,m\\}$ is reduced if in the first row and column the
integers occur in natural order. Latin square of side $m$ is equivalent to the
multiplication table (Cayley table) of a quasigroup on $m$ elements, if we
introduce a boundary row and a column. In our constructions, we use symmetric
Latin squares instead of Steiner 2-designs.

We only consider undirected graphs without loops or multiple edges. For all
necessary information about graphs we refer to [3]. For a survey of designs
and Latin squares, we refer to [5]

##  2 First construction

Let $\mathcal{D}_{1},\dots,\mathcal{D}_{m}$ be arbitrary affine designs all
with parameters $(q,q^{d-2})$, where $m=(q^{d}-1)/(q-1)$ is the number of
parallel classes of blocks in each $\mathcal{D}_{i}$. For all $i\in[m]$, let
$\mathcal{D}_{i}=(\mathcal{P}_{i},\mathcal{B}_{i})$. Parallel classes in each
$\mathcal{D}_{i}$ are enumerated by integers from $[m]$ and $j$-th parallel
class of $\mathcal{D}_{i}$ is denoted by $\mathcal{B}_{i}^{j}$. For any
$x\in\mathcal{P}_{i}$, the block in the parallel class $\mathcal{B}_{i}^{j}$
which contains $x$ is denoted by $B_{i}^{j}(x)$.

Let $\mathcal{L}=(e(i,j))$ be a symmetric Latin square of side $m$ on the
integers set $[m]$.

For every pair $i,j$ choose an arbitrary bijection

$\sigma_{i,j}:\mathcal{B}_{i}^{e(i,j)}\rightarrow\mathcal{B}_{j}^{e(j,i)}.$

We require that $\sigma_{i,j}=\sigma_{j,i}^{-1}$.

###### Construction 1

Let $\Gamma$ be a graph defined as follows:

  * 1.

The vertex set of $\Gamma$ is $\displaystyle
V=\bigcup_{i=1}^{m}\mathcal{P}_{i}.$

  * 2.

Two different vertices $x\in\mathcal{P}_{i}$ and $y\in\mathcal{P}_{j}$ are
adjacent in $\Gamma$ if and only if

$y\notin\sigma_{ij}(B_{i}^{e(i,j)}(x))\quad\mathrm{for\,all}\quad i,j\in[m].$

###### Theorem 1

If $\Gamma$ is a graph from Construction 1, then $\Gamma$ is a divisible
design graph with parameters

$v=q^{d}(q^{d}-1)/(q-1),\quad k=q^{d-1}(q^{d}-1),$
$\lambda_{1}=q^{d-1}(q^{d}-q^{d-1}-1),\quad\lambda_{2}=q^{d-2}(q-1)(q^{d}-1),$
$m=(q^{d}-1)/(q-1),\quad n=q^{d}.$

Moreover, $\Gamma$ has four distinct eigenvalues

$\\{q^{d}(q^{d-1}-1),\,q^{d-1},\,0,\,-q^{d-1}\\}.$

Proof.  Let $\Gamma$ be a graph from Construction 1. The number of vertices
$v$ is equal to $q^{d}m=q^{d}(q^{d}-1)/(q-1)$.

If $x$ is a vertex of $\Gamma$ belonging to $\mathcal{P}_{i}$, then

$\Gamma(x)=\bigcup_{j=1}^{m}(\mathcal{P}_{j}\setminus\sigma_{ij}(B_{i}^{e(i,j)}(x))).$

Clearly,

$|\mathcal{P}_{j}\setminus\sigma_{ij}(B_{i}^{e(i,j)}(x)|=|\mathcal{P}_{j}|-|\sigma_{ij}(B_{i}^{e(i,j)}(x)|=q^{d}-q^{d-1}.$

Hence, $\Gamma$ is a regular graph of degree

$k=(q^{d}-q^{d-1})(q^{d}-1)/(q-1)=q^{d-1}(q^{d}-1).$

Let $x$ and $y$ be two different vertices in $\Gamma$, belonging to the same
class $\mathcal{P}_{i}$. There are exactly $\lambda=(q^{d-1}-1)/(q-1)$ blocks
in $\mathcal{D}_{i}$ containing $x$ and $y$. Thus, there are
$(q^{d-1}-1)/(q-1)$ classes $\mathcal{P}_{j}$ in which
$\sigma_{ij}(B_{i}^{e(i,j)}(x))$ and $\sigma_{ij}(B_{i}^{e(i,j)}(y))$ are the
same blocks. Hence, $x$ and $y$ have exactly $q^{d}-q^{d-1}$ common neighbours
in each of these classes. In the remaining classes
$\sigma_{ij}(B_{i}^{e(i,j)}(x))$ and $\sigma_{ij}(B_{i}^{e(i,j)}(y))$ are
disjoint blocks. Hence, $x$ and $y$ have exactly $q^{d}-2q^{d-1}$ common
neighbours in each of the remaining classes. Therefore, the number of common
neighbours for $x$ and $y$ equals

$(q^{d}-q^{d-1})\frac{(q^{d-1}-1)}{(q-1)}+(q^{d}-2q^{d-1})(\frac{(q^{d}-1)}{(q-1)}-\frac{(q^{d-1}-1)}{(q-1)})=$
$=q^{d-1}(q^{d}-q^{d-1}-1).$

Let $x$ be in $\mathcal{P}_{i}$, and $y$ be in $\mathcal{P}_{j}$, where $i\neq
j$. In this case, for each $h\in[m]$ there are two blocks

$\sigma_{ih}(B_{i}^{e(i,h)}(x))\quad\mathrm{and}\quad\sigma_{jh}(B_{j}^{e(j,h)}(y))$

in $\mathcal{D}_{h}$ which have exactly $q^{d-2}$ points in common. Thus, $x$
and $y$ have exactly $q^{d}-2q^{d-1}+q^{d-2}$ common neighbours in each of
$m=(q^{d}-1)/(q-1)$ classes. Hence, the number of common neighbours for $x$
and $y$ equals

$(q^{d}-2q^{d-1}+q^{d-2})\frac{(q^{d}-1)}{(q-1)}=q^{d-2}(q-1)(q^{d}-1).$

$\square$

Remark that J. Guo, K. Wang and F. Li in [9, Theorem 3.3] obtained divisible
design graphs based on symplectic spaces with parameters

$\frac{q^{\nu+s}(q^{\nu-s}-1)}{(q-1)},q^{\nu+s-1}(q^{\nu-s}-1),q^{\nu+s-2}(q^{\nu-s}-1)(q-1),q^{\nu+s-1}(q^{\nu-s}-q^{\nu-s}-1).$

These parameters coincide with the parameters in Theorem 1 whenever $s=0$.

We keep all the notation introduced in this section until the end of the
article.

##  3 Second construction

Let $\Gamma^{\ast}$ be a divisible design graph from Construction 1 with
parameters
$(v^{\ast},k^{\ast},\lambda_{1}^{\ast},\lambda_{2}^{\ast},m^{\ast},n^{\ast})$
which was created using the Latin square
$\mathcal{L^{\ast}}=(\tau^{\ast}(i,j))$.

Let $\mathcal{L}=(e(i,j))$ be the square derived from the Latin square
$\mathcal{L}^{\ast}=(\tau^{\ast}(i,j))$ by removing the row $h$ and the column
$h$ for some $h$ from $[m^{\ast}]$. Since $\tau^{\ast}(i,h)$ and
$\tau^{\ast}(h,i)$ are absent in $\mathcal{L}$ we can use a sort of randomness
for $\mathcal{L}$ changing any $e(i,i)$ by $\tau^{\ast}(i,h)$ for $i<h$ and
$e(i,i)$ by $\tau^{\ast}(i+1,h)$ for $i\geq h$. Thus, for each $h$ there are
$2^{m^{\ast}-1}$ possibilities for choosing $\mathcal{L}$.

###### Construction 2

Let $\Gamma$ be a graph defined as follows:

  * 1.

The vertex set of $\Gamma$ is $\displaystyle
V=V^{\ast}\setminus\mathcal{P}_{h}.$

  * 2.

Two different vertices $x\in\mathcal{P}_{i}$ and $y\in\mathcal{P}_{j}$ are
adjacent in $\Gamma$ if and only if

$y\notin\sigma_{ij}(B_{i}^{e(i,j)}(x))\quad\mathrm{for\,all}\quad
i,j\in[m^{\ast}-1].$

###### Theorem 2

If $\Gamma$ is a graph from Construction 2, then $\Gamma$ is a divisible
design graph with parameters

$v=q^{d+1}(q^{d-1}-1)/(q-1),\quad k=q^{d}(q^{d-1}-1),$
$\lambda_{1}=q^{d}(q^{d-1}-q^{d-2}-1),\quad\lambda_{2}=q^{d-1}(q-1)(q^{d-1}-1),$
$m=q^{2}(q^{d-1}-1)/(q-1),\quad n=q^{d-1}.$

Moreover, $\Gamma$ has four distinct eigenvalues

$\\{q^{d}(q^{d-1}-1),\,q^{d-1},\,0,\,-q^{d-1}\\}.$

Proof.  Let $\Gamma$ be a graph from Construction 2. The number of vertices
$v$ is equal to

$v^{\ast}-q^{d}=q^{d+1}\frac{(q^{d-1}-1)}{(q-1)}.$

If $x$ is a vertex of $\Gamma$ belonging to $\mathcal{P}_{i}$, then there are
$k^{\ast}-(q^{d}-q^{d-1})$ vertices in $\Gamma(x)$. Hence, $\Gamma$ is a
regular graph of degree

$k=(q^{d}-q^{d-1})(m^{\ast}-1)=q^{d}(q^{d-1}-1).$

Let $x$ and $y$ be two different vertices from $\mathcal{D}_{i}$ belonging to
one block of $h$-th parallel class. Since $\mathcal{P}_{h}$ is not in $\Gamma$
there are $\lambda-1=(q^{d-1}-1)/(q-1)-1$ classes $\mathcal{P}_{j}$, where
$\sigma_{ij}(B_{i}^{e(i,j)}(x))$ and $\sigma_{ij}(B_{i}^{e(i,j)}(y))$ are the
same. Hence, the difference from $\lambda^{\ast}$ and $\lambda$ equals
$q^{d}-q^{d-1}$. Therefore, the number of common neighbours of $x$ and $y$ in
$\Gamma$ equals

$\lambda_{1}^{\ast}-(q^{d}-q^{d-1})=q^{d}(q^{d-1}-q^{d-2}-1).$

Let $x$ and $y$ don’t lie on one block of $h$-th parallel class in
$\mathcal{P}_{i}$ or $x\in\mathcal{P}_{i}$ and $y\in\mathcal{P}_{j}$, $i\neq
j$. In both cases, $x$ and $y$ have exactly $q^{d}-2q^{d-1}+q^{d-2}$ common
neighbours in each part of $V$. Hence, the number of common neighbours of $x$
and $y$ is equal to

$\lambda_{2}^{\ast}-(q^{d}-2q^{d-1}+q^{d-2})=q^{d-1}(q-1)(q^{d-1}r-1).$

Since $x$ and $y$ have the same number of common neighbours if and only if
they belong to one block in $h$-th parallel class in $\mathcal{P}_{i}$ for all
$i\in[m^{\ast}]\setminus\\{h\\}$, then $\Gamma$ is a divisible design graph
with classes on all these blocks. Therefore, $n=q^{d-1}$ and
$m=v/n=q^{2}(q^{d-1}-1)/(q-1)$. $\square$

##  4 Third construction

Let $\mathcal{D}_{1},\dots,\mathcal{D}_{m+1}$ be arbitrary affine designs all
with parameters $(q,q^{d-2})$, where $m=(q^{d}-1)/(q-1)$ is the number of
parallel classes of blocks in each $\mathcal{D}_{i}$. For all $i\in[m+1]$, let
$\mathcal{D}_{i}=(\mathcal{P}_{i},\mathcal{B}_{i})$. Parallel classes in each
$\mathcal{D}_{i}$ are enumerated by integers from $[m]$.

Let $\mathcal{L}=(e(i,j))$ be a symmetric Latin square of side $m+1$ on the
integers set $[m+1]$.

###### Construction 3

Let $\Gamma$ be a graph defined as follows:

  * 1.

The vertex set of $\Gamma$ is $\displaystyle
V=\bigcup_{i=1}^{m+1}\mathcal{P}_{i}.$

  * 2.

Two different vertices $x\in\mathcal{P}_{i}$ and $y\in\mathcal{P}_{j}$ are
adjacent in $\Gamma$ if and only if

$y\notin\sigma_{ij}(B_{i}^{e(i,j)}(x))\quad\mathrm{for\,all}\quad
i,j\in[m+1].$

If $e(i,j)=m+1$, then there are no edges between $\mathcal{P}_{i}$ and
$\mathcal{P}_{j}$.

###### Theorem 3

Let $\Gamma$ be a graph from Construction 3. If $q\neq 2$, then $\Gamma$ is a
divisible design graph with parameters

$v=q^{d}(q^{d}+q-2)/(q-1),\quad k=q^{d-1}(q^{d}-1),$
$\lambda_{1}=q^{d-1}(q^{d}-q^{d-1}-1),\quad\lambda_{2}=q^{d-1}(q-1)(q^{d-1}-1),$
$m=(q^{d}+q-2)/(q-1),\quad n=q^{d}.$

Moreover, $\Gamma$ has five distinct eigenvalues

$\\{q^{d-1}(q^{d}-1),\,(q-1)q^{d-1},\,q^{d-1},\,-q^{d-1},\,-(q-1)q^{d-1}\\}.$

If $q=2$, then $\Gamma$ is strongly regular graph with parameters

$(2^{2d},2^{2d-1}-2^{d-1},2^{2d-2}-2^{d-1},2^{2d-2}-2^{d-1}).$

Proof.  Let $\Gamma$ be a graph from Construction 3. The number of vertices
$v$ is equal to

$q^{d}(m+1)=q^{d}(q^{d}+q-2)/(q-1).$

If $x$ is a vertex of $\Gamma$ belonging to $\mathcal{P}_{i}$, then

$\Gamma(x)=\bigcup_{j=1,\,e(i,j)\neq
m+1}^{m+1}(\mathcal{P}_{j}\setminus\sigma_{ij}(B_{i}^{e(i,j)}(x))).$

Hence, $\Gamma$ is a regular graph of degree

$k=(q^{d}-q^{d-1})(q^{d}-1)/(q-1)=q^{d-1}(q^{d}-1).$

Let $x$ and $y$ be two different vertices belonging to $\mathcal{P}_{i}$.
Since there are no edges between $\mathcal{P}_{i}$ and $\mathcal{P}_{j}$ if
$e(i,j)=m+1$, the number of common neighbours for $x$ and $y$ equals
$q^{d-1}(q^{d}-q^{d-1}-1)$ exactly the same as in Theorem 1.

Let $x$ be in $\mathcal{P}_{i}$, and $y$ be in $\mathcal{P}_{j}$, where $i\neq
j$. In this case, $x$ and $y$ have exactly $q^{d}-2q^{d-1}+q^{d-2}$ common
neighbours in each class $\mathcal{P}_{h}$, where $e(i,h)\neq m+1$ or
$e(h,j)\neq m+1$. Hence, the number of common neighbours for $x$ and $y$
equals

$(q^{d}-2q^{d-1}+q^{d-2})\left(\frac{(q^{d}-1)}{(q-1)}-1\right)=q^{d-1}(q-1)(q^{d-1}-1).$

If $q=2$, then $\lambda_{1}=\lambda_{2}$ and $\Gamma$ is strongly regular
graph. $\square$

##  5 Fourth construction

Let $\mathcal{D}_{1},\dots,\mathcal{D}_{m+1}$ be arbitrary affine designs all
with parameters $(q,q^{d-2})$, where $m=(q^{d}-1)/(q-1)$ is the number of
parallel classes of blocks in each $\mathcal{D}_{i}$. For all $i\in[m+1]$, let
$\mathcal{D}_{i}=(\mathcal{P}_{i},\mathcal{B}_{i})$. Parallel classes in each
$\mathcal{D}_{i}$ are enumerated by integers from $[m]$.

Let $\mathcal{L}=(e(i,j))$ be a symmetric Latin square of side $m+1$ and there
are no entries $m+1$ on the main diagonal.

###### Construction 4

The graph $\Gamma$ is defined as follows:

  * 1.

The vertex set of $\Gamma$ is $\displaystyle
V=\bigcup_{i=1}^{m+1}\mathcal{P}_{i}.$

  * 2.

Two vertices $x\in\mathcal{P}_{i}$ and $y\in\mathcal{P}_{j}$ are adjacent in
$\Gamma$ if and only if

$y\notin\sigma_{ij}(B_{i}^{e(i,j)}(x))\quad\mathrm{for\,all}\quad
i,j\in[m+1].$

If $\ell(i,j)=m+1$, then every vertex from $\mathcal{P}_{i}$ is adjacent to
all vertices from $\mathcal{P}_{j}$.

###### Theorem 4

If $\Gamma$ is a graph from Construction 4, then $\Gamma$ is a strongly
regular graph with parameters $(v,k,\lambda,\mu)$, where

$v=q^{d}(q^{d}+q-2)/(q-1),\quad k=q^{d-1}(q^{d}+q-1),$
$\lambda=\mu=q^{d-1}(q-1)(q^{d-1}+1).$

Proof.  Let $\Gamma$ be a graph from Construction 4. If $x$ is a vertex from
$\mathcal{P}_{i}$, then there exists the only $h\in[m+1]$ such that
$e(i,j)=m+1$. Hence,

$\Gamma(x)=\mathcal{P}_{h}\cup\bigcup_{j=1,\,j\neq
h}^{m+1}(\mathcal{P}_{j}\setminus\sigma_{ij}(B_{i}^{e(i,j)}(x))).$

Therefore, $\Gamma$ is a regular graph of degree

$k=q^{d}+(q^{d}-q^{d-1})(q^{d}-1)/(q-1)=q^{d-1}(q^{d}+q-1).$

Let $x$ and $y$ be two different vertices in $\Gamma$, belonging to the same
class $\mathcal{P}_{i}$. Since every vertex from $\mathcal{P}_{i}$ is adjacent
to all vertices from $\mathcal{P}_{j}$ in the case $e(i,j)=m+1$ then the
number of common neighbours for $x$ and $y$ equals

$q^{d}+q^{d-1}(q^{d}-q^{d-1}-1)=q^{d-1}(q-1)(q^{d-1}+1).$

Let $x$ be in $\mathcal{P}_{i}$, and $y$ be in $\mathcal{P}_{j}$, where $i\neq
j$. In this case, $x$ and $y$ have exactly $q^{d}-2q^{d-1}+q^{d-2}$ common
neighbours in each class $\mathcal{P}_{h}$, where $e(i,h)\neq m+1$ or
$e(h,j)\neq m+1$ and $q^{d}-q^{d-1}$ common neighbours in $\mathcal{P}_{h}$,
where $e(i,h)=m+1$ or $e(h,j)=m+1$. Hence, the number of common neighbours for
$x$ and $y$ equals

$2(q^{d}-q^{d-1})+((q^{d}-2q^{d-1}+q^{d-2})\left(\frac{(q^{d}-1)}{(q-1)}-1\right)=$
$=q^{d-1}(q-1)(q^{d-1}+1).$

Thus, $\Gamma$ is strongly regular graph. $\square$

###### Remark 1

The complement of a graph from Construction 4 is a strongly regular graph with
parameters

$(q^{d}\frac{(q^{d}+q-2)}{(q-1)},\,q^{d-1}\frac{(q^{d}-1))}{(q-1)}-1,\,q^{d-1}\frac{(q^{d-1}-1)}{(q-1)}-2,\,q^{d-1}\frac{(q^{d-1}-1)}{(q-1)}).$

These parameters were first obtained in Construction B of type $S6$ in the
paper [15] by M. Muzychuk. We were unable to check if the complement of any
graph from Construction 4 can be obtained from Construction S6.

##  6 Hadamard matrices and p-rank

Let $J$ is the all ones matrix and $I$ is the identity matrix.

A square $(+1,-1)$-matrix $H$ is called a Hadamard matrix of order $n$
whenever $HH^{T}=nI$. A Hadamard matrix $H$ is called graphical if $H$ is
symmetric and it has constant diagonal, and $H$ is regular if all row and
column sums are equal.

In Section 4 of the paper [1] we find the following statement. If $H$ is a
graphical regular Hadamard matrix with the diagonal entries are equal to $1$,
then $\cfrac{1}{2}(J-H)$ is the adjacency matrix of a strongly regular graph
with parameters

$(n,\,\cfrac{n}{2}\mp\cfrac{\sqrt{n}}{2},\,\cfrac{n}{4}\mp\cfrac{\sqrt{n}}{2},\,\cfrac{n}{4}\mp\cfrac{\sqrt{n}}{2}).$

Conversely, any strongly regular graph with one of two these tuples of
parameters comes from a Hadamard matrix in the described way.

If $n=2^{2d}$, then strongly regular graphs have parameters

$(2^{2d},2^{2d-1}\mp 2^{d-1},2^{2d-2}\mp 2^{d-1},2^{2d-2}\mp 2^{d-1}).$

Thus, Theorem 3 and Theorem 4 are proposed as another point of view on these
graphs in the even case.

One of the well-studied and useful parameters of a graph is the $p$-rank, that
is, the rank of its adjacency matrix over the field with $p$-elements [4]. The
$p$-rank can be useful to distinguish divisible design graphs and strongly
regular graphs with the same parameters and spectrum [1].

Variation of the numbering of parallel classes of blocks in the designs for
all our constructions does not change the parameters and spectrum of the
adjacency matrix of the resulting graph. Nevertheless, such variation can
change the $2$-rank, in which case the new graph is obviously not isomorphic
to the original one.

##  7 Examples of divisible design graphs from Construction 1, 2 and 3

All feasible parameter and known divisible design graphs with at most 27
vertices are given in [6, 10]. Divisible design graphs were studied in
master’s thesis by M.A. Meulenberg in [14], where one can find feasible
parameters of such graphs up to 50 vertices. All divisible design graphs with
at most 39 vertices, except for three tuples of parameters: $(32,15,6,7,4,8)$,
$(32,17,8,9,4,8)$, $(36,24,15,16,4,9)$ were found by D.I. Panasenko and L.V.
Shalaginov in [16]. Due to the large number of examples in this and the
following sections, we will only focus on examples derived from reduced Latin
squares.

###  7.1 q=2, d=2

By Theorem 1 and Theorem 2 divisible design graphs have parameters
$(12,6,2,3,3,4)$ and $(8,4,0,2,4,2)$, respectively. Only line graph of the
octahedron has the first one, and $K_{4}\times K_{2}$ has the second one.

###  7.2 q=3, d=2

By Theorem 1 and Theorem 2, parameters of divisible design graphs are
$(36,24,15,16,4,9)$ and $(27,18,9,12,18,3)$, respectively. There are two
symmetric Latin squares of side $4$, which are the multiplication tables for
$C_{4}$ the cyclic group of order $4$ and the Klein four-group. These Latin
squares give two divisible design graphs with parameters $(36,24,15,16,4,9)$.
Both of these graphs are known as Cayley graphs from [12, Example 4.2].

There are two known distant-regular graphs with intersection array
$\\{8,6,1;1,3,8\\}$. The complement of these graphs are divisible design
graphs with parameters $(27,18,9,12,18,3)$. We can obtain both of these graphs
using Theorem 2 as well.

###  7.3 q=2, d=3

By Theorem 1 divisible design graphs have parameters $(56,28,14,12,7,8)$.
There are seven symmetric Latin squares of side $7$ (see for example Table
1.18 in Part III [5]). We use $7$ copies of a $3$-dimensional point-hyperplane
affine design over a finite field of order $2$ and identity for all
bijections. For each parallel class of blocks take adjacency matrix of the
complete multipartite graph on the vertex set of the design with parts on
parallel blocks. We get seven matrices enumerated by parallel classes.
Changing any entries of Latin squares by the matrix with the number equals of
the entry we have got seven adjacency matrices of divisible design graphs with
parameters $(56,28,14,12,7,8)$. We used the computer system GAP with Package
GRAPE [8, 17] to found these graphs.

Divisible design graphs from the second and fourth Latin squares, as well as
the sixth and seventh one are isomorphic. Hence, we obtained $5$ non-
isomorphic divisible design graphs with parameters $(56,28,14,12,7,8)$.

Any variation of the numbering of parallel classes in the starting design
often gives non-isomorphic graphs.

By Theorem 2 we have parameters $(48,24,8,12,12,4)$ and obtained lots of non-
isomorphic divisible design graphs with such parameters.

By Theorem 3 we have parameters $(45,24,15,12,5,9)$. There are one symmetric
Latin square of side $5$ (see for example Table 1.18 in Part III [5]). The
graph we obtained using this Latin square from Construction 3 has the
automorphism group $(C_{3}\times C_{3}):C_{8}$.

All graphs with parameters $(56,28,14,12,7,8)$, $(48,24,8,12,12,4)$ and
$(45,24,15,12,5,9)$ seem to be new and are available on the website [19].

##  8 Examples of strongly regular graphs from Construction 3 and 4

###  8.1 q=2, d=3

A. Abiad, S. Butler and W.H. Haemers in [1] and F. Ihringer [11] provided an
abundance of strongly regular graphs with parameters $(64,28,12,12)$. In the
libraries of small loops in GAP package LOOPS [13] we found symmetric Latin
squares as multiplication tables of abelian loops of order 8. We used the
multiplication tables of three abelian groups of order $8$ as well. Using
these Latin squares we obtained fifteen non-isomorphic graphs with parameters
$(64,28,12,12)$ from Construction 3. The following table is contain the
results, where the first column gives the $2$-rank, and the second column
gives the structure description of the automorphism group.

8 |  $(C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}):S_{8}$;
---|---
8 |  $((((((C_{2}\times C_{2}\times(((C_{2}\times C_{2}\times C_{2}\times C_{2}):C_{2}):C_{2})):C_{2}):C_{2}):C_{2}):C_{2}):C_{3}):C_{2}$;
10 |  $C_{2}\times C_{2}\times C_{2}\times D_{8}$;
10 |  $C_{2}\times C_{2}\times(((C_{2}\times C_{2}\times C_{2}\times C_{2}):C_{3}):C_{2})$;
10 |  $C_{2}\times(((((C_{2}\times((C_{2}\times C_{2}\times C_{2}\times C_{2}):C_{2})):C_{2}):C_{3}):C_{2}):C_{2})$;
12 |  $C_{2}\times C_{2}\times C_{2}\times D_{8}$;
12 |  $C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}$;
12 |  $C_{2}\times D_{8}\times D_{8}$;
12 |  $C_{2}\times((C_{2}\times((C_{4}\times C_{2}):C_{2})):C_{2})$;
12 |  $C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}$;
12 |  $C_{2}\times C_{2}\times C_{2}\times D_{8}$;
12 |  $(C_{2}\times C_{2}\times C_{2}\times D_{8}):C_{2}$;
12 |  $(C_{4}\times C_{2}\times C_{2}\times C_{2}\times C_{2}):(C_{2}\times A_{5})$;
14 |  $C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}$;
14 |  $C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}$.

We have one graph with the automorphism group of order $7680$ and the
structure description $(C_{4}\times C_{2}\times C_{2}\times C_{2}\times
C_{2}):(C_{2}\times A_{5})$. This graph was created using one of conjugacy
closed loops from the CC library of GAP package LOOPS [13]. Such a graph was
not found in [11].

###  8.2 q=3, d=3

Strongly regular graphs with parameters $(378,261,180,180)$ were first
obtained from Construction B of type $S6$ in the paper [15] by M. Muzychuk. By
Construction 4 we found three strongly regular graphs with such parameters.
There are three symmetric Latin square of side $14$. First one is the Cayley
table of the cyclic group $C_{14}$ and two others are the Cayley table of
Steiner loops in the library of GAP package LOOPS [13]. Using this Latin
squares we obtained three non-isomorphic strongly regular graphs with
parameters $(378,261,180,180)$. Their $3$-ranks and the structural
descriptions of the automorphism groups are given in the following table.

66 |  $(C_{3}\times C_{3}\times C_{3}):C_{2}$;
---|---
65 |  $C_{2}\times((C_{3}\times C_{3}\times C_{3}):C_{2})$;
65 |  $((A_{9}\times A_{9}\times A_{9}):(C_{2}\times S_{4})$.

All graphs with parameters $(64,28,12,12)$ and $(378,261,180,180)$ which were
found in this section are available on the website [19].

## References

  * [1] A. Abiad, S. Butler and W.H. Haemers, Graph switching, 2-ranks, and graphical Hadamard matrices, Discrete Mathematics, 342 (2019) 2850–2855. https://doi.org/10.1016/J.DISC.2018.11.022
  * [2] R.C. Bose, W.S. Connor, Combinatorial properties of group divisible incomplete block designs, Ann. Math. Statist., 23 (1952) 367–383. https://doi.org/10.1214/AOMS/1177729382
  * [3] A.E. Brouwer, W.H. Haemers, Spectra of graphs, in: Springer Universitext, 2012.
  * [4] A.E. Brouwer, C.A. van Eijl, On the p-rank of the adjacency matrix of strongly regular graphs, J. Alg. Combin. 1 (1992) 329–346.
  * [5] Ch.J. Colbourn, J.H. Dinitz, I.M. Wanless, Latin Squares, In Handbook of Combinatorial Designs (2nd ed.) (2007) 135–151.
  * [6] D. Crnković, W. H. Haemers, Walk-regular divisible design graphs, Designs, Codes and Cryptography, 72 (2014) 165–175. https://doi.org/10.1007/s10623-013-9861-0
  * [7] D.G. Fon-Der-Flaass New prolific constructions of strongly regular graphs,Adv. Geom., 2 (2002), 301–306.
  * [8] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.11.1; 2021. https://www.gap-system.org
  * [9] J. Guo, K. Wang, F. Li, Deza graphs based on symplectic spaces, European J. Combin., 31(8) (2010) 1969–1980. https://doi.org/10.1016/j.ejc.2010.05.006
  * [10] W.H. Haemers, H. Kharaghani, M. Meulenberg, Divisible design graphs, J. Combinatorial Theory, Series A, 118 (2011) 978–992. https://doi.org/10.1016/j.jcta.2010.10.003
  * [11] F. Ihringer, Switching for Small Strongly Regular Graphs, arXiv:2012.08390 [math.CO].
  * [12] V.V. Kabanov, L.V. Shalaginov, On divisible design Cayley graphs, The Art of Discrete and Applied Mathematics, 4 (2021) #P2.02. https://doi.org/10.26493/2590-9770.1340.364
  * [13] G. Nagy, P. Vojtěchovský, The LOOPS Package: Loops and quasigroups for GAP. https://gap-packages.github.io/loops/
  * [14] M.A. Meulenberg, Divisible Design Graphs, Master’s thesis, _Tilburg University_ (2008).
  * [15] M. Muzychuk, A generalization of Wallis-Fon-Der-Flaass construction of strongly regular graphs, J. Algebr. Comb., 25 (2007) 169–187. https://doi.org/10.1007/s10801-006-0030-7
  * [16] D.I. Panasenko, L.V. Shalaginov, Classification of divisible design graphs with at most 39 vertices, arXiv:2109.12805 [math.CO].
  * [17] L.H. Soicher, The GRAPE package for GAP, Version 4.8.5, 2021. https://gap-packages.github.io/grape
  * [18] W.D. Wallis, Construction of strongly regular graphs using affine designs, Bull. Austral. Math. Soc., 4 (1971), 41–49. https://doi.org/10.1017/S0004972700046244
  * [19] D.I. Panasenko, L.V. Shalaginov, Deza graphs http://alg.imm.uran.ru/dezagraphs/ddg.php