# Fractional revival on abelian Cayley graphs

Xiwang Cao, Gaojun Luo  Xiwang Cao is with the School of Mathematical
Sciences, Nanjing University of Aeronautics and Astronautics, Nanjing 210016,
China, and he is also Key Laboratory of Mathematical Modeling and High
Performance Computing of Air Vehicles(NUAA), MIIT, Nanjing 210016, China,
email<EMAIL_ADDRESS>Gaojun Luo is with School of Physical and
Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link,
Singapore 637371, Singapore, email<EMAIL_ADDRESS>
###### Abstract.

Fractional revival, known as a quantum transport phenomenon, is essential for
entanglement generation in quantum spin networks. The concept of fractional
revival is a generalization of perfect state transfer and periodicity on
graphs. In this paper, we propose a sufficient and necessary condition for
abelian Cayley graphs having fractional revival between any two distinct
vertices. With this characterization, two general constructions of abelian
Cayley graphs having fractional revival is presented. Meanwhile, we establish
several new families of abelian Cayley graphs admitting fractional revival.

MSC 2010: 05C25, 81P45, 81Q35.

###### Key words and phrases:

fractional revival, Cayley graph, plateaued function, cublike graph

Xiwang Cao has supported by National Natural Science Foundation of China under
grant numbers 11771007, 12171241,

##  1\. Introduction

In quantum information, continuous-time quantum walk is an essential
ingredient in transport problems of quantum spin networks. The study of
perfect state transfer in quantum spin chains dates back to [1]. Further
research has been established in [2, 3, 4]. In [5], Kay provided a
comprehensive survey on this topic. With entanglement generation in quantum
spin chains, the transport problem is said to be fractional revival in [6].
Investigating fractional revival is both of theoretical and practical
significance as it generalizes two quantum transport phenomena, namely,
perfect state transfer and periodicity.

Let $\Gamma=(V,E)$ be a connected simple graph with set $V$ of $n$ vertices
and set $E$ of edges. Let $A=A(\Gamma)=(a_{u,v})_{u,v\in V}$ be the adjacency
matrix of $\Gamma$ defined by

$a_{u,v}=\left\\{\begin{array}[]{cc}1,&\mbox{ if $(u,v)\in E$},\\\ 0,&\mbox{
otherwise}.\end{array}\right.$

The matrix $A$ is an $n\times n$ symmetric $\\{0,1\\}$-matrix and all
eigenvalues of $A$ are real numbers. The transfer matrix of $\Gamma$ is
defined by the following $n\times n$ unitary matrix

$H(t)=\exp(\imath tA)=\sum_{s=0}^{+\infty}\frac{(\imath
tA)^{s}}{s!}=(H_{u,v}(t))_{u,v\in V},\ t\in\mathbb{R},\imath=\sqrt{-1},$

where $\mathbb{R}$ is the set of real numbers.

######  Definition 1.1.

For two distinct vertices $x$ and $y$ of $\Gamma$, if

$H(t)\mathbf{e}_{x}=\alpha\mathbf{e}_{x}+\beta\mathbf{e}_{y}$ (1.1)

for some complex numbers $\alpha$ and $\beta$ satisfying $\beta\neq 0$ and
$|\alpha|^{2}+|\beta|^{2}=1$, then the graph $\Gamma$ admits
$(\alpha,\beta)$-fractional revival (or fractional revival, in short, FR) from
$x$ to $y$ at time $t$, where $\mathbf{e}_{x}$ is the standard vector whose
$x$-th entry is $1$ and $0$ otherwhere. Factoring a common unimodular phase
factor, $(\alpha,\beta)$-fractional revival is turned into
$e^{\imath\theta}(\alpha,\beta)$-fractional revival, where $\alpha$ and
$\theta$ are real scalars and $\beta$ is complex. If $\alpha=0$, then
$(0,\beta)$-fractional revival is reduced to perfect state transfer (in short,
PST). The graph $\Gamma$ is said to be periodic at some time $t$ if $\beta=0$
in (1.1).

The basic problem on this topic is to find graphs having FR. More
specifically, for a given connected simple graph $\Gamma=(V,E)$, and $u,v\in
V$, define

$\displaystyle T(u,v)$ $\displaystyle=$
$\displaystyle\\{t\in\mathbb{R}:t>0,H(t)\mathbf{e}_{u}=\alpha\mathbf{e}_{u}+\beta\mathbf{e}_{v}\\}.$

We want to determine $T(u,v)$ for any pair $\\{u,v\\}$ of vertices, or at
least, to determine if $\Gamma$ has FR, that is, $T(u,v)$ is not empty for
some pair $\\{u,v\\}$. PST on various graphs, which is a special case of FR,
has been investigated by many researchers. The works in [7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20] and the references therein form a partial list
of an already vast literature. With regard to PST, only a few attempts have
been made to FR. Genest, Vinet and Zhedanov in [6] systematically studied
quantum spin chains with FR at two sites. Later in [21], a graph exemplified
the existence of FR between antipodes. Chan et al. in [22] established a
general approach to constructing graphs exhibiting FR between cospectral
vertices. In addition, they characterized cycles and paths having FR. In a
consecutive paper [23], the same authors of [22] discussed FR on graphs with
adjacency matrices from the Bose-Mesner algebra of association schemes. A
comprehensive study of FR on graphs between cospectral and strongly cospectral
vertices was presented in [24]. Unlike the work in [24], Godsil and Zhang in
[25] proposed a class of graphs admitting FR between non-cospectral vertices.
Given a fixed element $a$ of an abelian group $G$, Wang, Wang and Liu in [26]
gave a sufficient and necessary condition for Cayley graphs on $G$ having FR
between vertices $x$ and $x+a$. The characterization in [26] is incomplete as
the element $a$ is fixed. Up to now, there is no sufficient and necessary
condition for Cayley graphs on abelian groups admitting FR between any two
distinct vertices.

This paper aims at characterizing FR on Cayley graphs over abelian groups
between any two distinct vertices. Our contribution begins by Theorem 2.2 in
the next section with a sufficient and necessary condition for abelian Cayley
graphs having FR between any two distinct vertices. Based on such a criteria,
we propose two general constructions of abelian Cayley graphs admitting FR in
Corollaries 2.6 and 2.10. As a consequence, two families of abelian Cayley
graphs admitting FR are provided in respective Propositions 2.7 and 2.11.
Furthermore, Propositions 2.7 and 2.11 tell us that the pool of feasible
graphs having FR is larger than that of graphs admitting PST. In Section 3, we
propose the definition of plateaued functions over abelian groups. With the
help of these functions, a family of abelian Cayley graphs having FR is built
in Theorem 3.7. In Section 4, we discuss FR on cublike graphs. A flexible
construction of cublike graphs admitting FR is given in Theorem 4.1. Utilizing
bent or semi-bent functions over finite fields of even characteristics, we
present the last construction of cublike graphs having FR in Theorem 4.5.
Section 5 concludes this paper.

##  2\. A characterization of abelian Cayley graphs having FR

Let $G$ be an abelian group of order $n$ and let $S$ be a subset of $G$ with
$S=-S$ and $G=\langle S\rangle$. Let $\mathbb{C}$, $\mathbb{Z}$ and
$\mathbb{Q}$ be, respectively, the set of complex numbers, the set of integers
and the set of rational numbers. Let $\Gamma={\rm Cay}(G,S)$ be an abelian
Cayley graph whose vertex set is $G$ and two vertices, $u,v$, form an edge of
$\Gamma$ if $u-v\in S$. The adjacency matrix of $\Gamma$ is an $n\times n$
matrix $A=(a_{g,h})$ defined by

$a_{g,h}=\left\\{\begin{array}[]{cc}1,&\mbox{ if $g-h\in S$},\\\ 0,&\mbox{
otherwise}.\end{array}\right.$

An automorphism of $\Gamma$ is a bijection from $G$ to $G$ preserving the
adjacency. The set of automorphism of $\Gamma$ is denoted by ${\rm
Aut}(\Gamma)$. For every $z\in G$, define $\sigma_{z}:G\rightarrow G;x\mapsto
x+z$. It is easy to see that $\sigma_{z}\in{\rm Aut}(\Gamma)$ and $G$ is a
subgroup of ${\rm Aut}(\Gamma)$.

Each finite abelian group $G$ can be decomposed as a direct sum of cyclic
groups:

$G=\mathbb{Z}_{n_{1}}\oplus\cdots\oplus\mathbb{Z}_{n_{r}},\ \ n_{s}\geq 2,$

where $\mathbb{Z}_{m}=(\mathbb{Z}/m\mathbb{Z},+)$ is a cyclic group of order
$m$. For every $x=(x_{1},\cdots,x_{r})\in G$ with
$x_{s}\in\mathbb{Z}_{n_{s}}$, the mapping

$\chi_{x}:G\rightarrow\mathbb{C},\
\chi_{x}(g)=\prod_{s=1}^{r}\omega_{n_{s}}^{x_{s}g_{s}},\
g=(g_{1},\cdots,g_{r})\in G$

is a character of $G$, where $\omega_{n_{s}}=\exp(2\pi i/n_{s})$ is a
primitive $n_{s}$-th root of unity in $\mathbb{C}$. The set of characters of
$G$, denoted by $\hat{G}$, forms a group under the following operation:

$\chi_{g}\chi_{h}:G\rightarrow\mathbb{C};x\mapsto\chi_{g}(x)\chi_{h}(x),\forall
g,h\in G.$

Let $\hat{G}$ be the character group of $G$. Recall that for any abelian group
$G$, $G$ is isomorphic to its dual group via
$G\rightarrow\hat{G},g\mapsto\chi_{g}$. It is known that the eigenvalues of
$A$ are the real numbers

$\lambda_{g}=\sum_{s\in S}\chi_{g}(s),\forall g\in{G}.$

Let $P=\frac{1}{\sqrt{n}}(\chi_{g}(h))_{g,h\in G}$ and $E_{z}=p_{z}^{*}p_{z}$
with $p_{z}$ being the $z$-th column of $P$ and $p_{z}^{*}$ being the
conjugate transpose of $p_{z}$. The spectral decomposition of $A$ is given by

$A=\sum_{g\in G}\lambda_{g}E_{g}.$ (2.1)

Since $\\{E_{z}:z\in G\\}$ forms a set of idempotents, namely,

$E_{g}E_{h}=\left\\{\begin{array}[]{cc}E_{g},&\mbox{ if $g=h$},\\\
0,&\mbox{otherwise},\end{array}\right.$

we have the spectral decomposition of $H(t)$ as follows:

$H(t)=\sum_{g\in G}e^{\imath\lambda_{g}t}E_{g}.$ (2.2)

Thus the $(g,h)$-th entry of $H(t)$ is

$H(t)_{g,h}=\frac{1}{n}\sum_{z\in G}e^{\imath t\lambda_{z}}\chi_{z}(g-h).$
(2.3)

If FR happens on $\Gamma={\rm Cay}(G,S)$ from a vertex $x$ to a vertex $y$,
then by (1.1) and noticing that $H(t)$ is a unitary matrix, the $x$-th row of
$H(t)$ is determined by

$H(t)_{x,u}=\left\\{\begin{array}[]{cc}\alpha,&\mbox{ if $u=x$},\\\
\beta,&\mbox{ if $u=y$},\\\ 0,&\mbox{otherwise}.\end{array}\right.$ (2.4)

Since $A$ is an $G$-invariant matrix, namely, $a_{g+z,h+z}=a_{g,h},\forall
g,h,z\in G$, we know that $H(t)$ is also $G$-invariant. Therefore, we have the
following preliminary result.

######  Lemma 2.1.

Let $G$ be an abelian group of order $n$ and let $S$ be a subset of $G$. Let
$\Gamma={\rm Cay}(G,S)$ be a Cayley graph with the adjacency matrix $A$. Then
$\Gamma$ has $(\alpha,\beta)$-FR from a vertex $x$ to a vertex $y$ at some
time $t$ if and only if

$H(t)=\alpha I_{n}+\beta Q,$ (2.5)

where $Q$ is a permutation matrix satisfying $Q^{T}=Q$ and $Q^{2}=I_{n}$, the
identity matrix of order $n$. Moreover, $Q$ has no fixed points and
$\alpha\overline{\beta}+\overline{\alpha}\beta=0$.

###### Proof.

Suppose that $H(t)=\alpha I_{n}+\beta Q$ for some $t$. By the fact that $Q$ is
a permutation matrix with no fixed points, we can assume that there is some
$a\neq 0$ such that

$Q_{0,a}=1,Q_{0,b}=0,\forall b\neq 0.$

Then we have

$H(t)_{0,0}=\alpha,H(t)_{0,a}=\beta,\mbox{ and }H(t)_{0,b}=0,\forall b\neq
0,a.$

Therefore the graph $\Gamma$ exhibits $(\alpha,\beta)$-FR from $0$ to $a$.

Conversely, if $\Gamma$ has $(\alpha,\beta)$-FR from $x$ to $y$, then

$H(t)\mathbf{e}_{x}=\alpha\mathbf{e}_{x}+\beta\mathbf{e}_{y},$

which implies that

$H(t)_{x,x}=\alpha,H(t)_{x,y}=\beta\mbox{ and }H(t)_{x,b}=0,\forall b\neq
x,y.$

Since $H(t)$ is $G$-invariant, we have that for every $z\in G$,

$H(t)_{x+z,x+z}=\alpha,H(t)_{x+z,y+z}=\beta\mbox{ and
}H(t)_{x+z,b+z}=0,\forall b\neq x,y.$

When $z$ runs through $G$, we get that $H(t)=\alpha I_{n}+\beta Q$, where $Q$
is a permutation matrix with no fixed points. The matrix $Q$ is symmetric as
$H(t)$ is a symmetric matrix. Since $H(t)$ is a unitary matrix, we obtain that

$H(t)\overline{H(t)}=I_{n}\Leftrightarrow\alpha\overline{\alpha}I_{n}+\beta\overline{\beta}Q^{2}+(\alpha\overline{\beta}+\overline{\alpha}\beta)Q=I_{n}.$

Therefore, we have

$\left\\{\begin{array}[]{l}Q^{2}=I_{n},\\\
\alpha\overline{\beta}+\overline{\alpha}\beta=0.\end{array}\right.$

This completes the proof. ∎

From now on, we only consider the case of $(\alpha,\beta)$-FR with $\alpha\neq
0$. That is, we always assume that $\alpha\beta\neq 0$.

Recall that $H(t)$ is $G$-invariant, if FR happens on $\Gamma={\rm Cay}(G,S)$
from a vertex $x$ to a vertex $y$, then we have $H(t)_{x,u}=H(t)_{0,u-x}$,
thus we can write

$H(t)_{0,u}=\frac{1}{n}\sum_{z\in G}e^{\imath
t\lambda_{z}}{\chi_{z}(u)}=\left\\{\begin{array}[]{cc}\alpha,&\mbox{ if
$u=0$},\\\ \beta,&\mbox{ if $u=y-x$},\\\
0,&\mbox{otherwise}.\end{array}\right.$ (2.6)

Denote $a=x-y$. According to (2.6) and its Fourier inversion, we get

$e^{\imath\lambda_{g}t}=\alpha+\overline{\chi_{a}(g)}\beta,\forall g\in G.$
(2.7)

Since $\lambda_{g}$ is real for every $g\in G$, we derive that

$1=e^{\imath\lambda_{g}t}\overline{e^{\imath\lambda_{g}t}}=(\alpha+\overline{\chi_{a}(g)}\beta)(\overline{\alpha}+{\chi_{a}(g)}\overline{\beta})=1+\overline{\chi_{a}(g)}\overline{\alpha}\beta+{\chi_{a}(g)}\alpha\overline{\beta},$

which is equivalent to

$\chi_{a}(g){\alpha}\overline{\beta}+\overline{\chi_{a}(g)}\overline{\alpha}{\beta}=0.$
(2.8)

Combing with $\overline{\alpha}\beta+\alpha\overline{\beta}=0$ in Lemma 2.1,
we have

$\chi_{a}(g)=\overline{\chi_{a}(g)},\mbox{ i.e., }\chi_{a}(g)=\pm 1,\forall
g\in G.$ (2.9)

That is, $a$ is of order two in $G$. As a consequence, FR cannot happen on any
abelian Cayley graph with the underlying group has an odd order.

Define two subsets of $G$ as follows,

$G_{0}=\\{g\in G:\chi_{a}(g)=1\\},G_{1}=\\{g\in G:\chi_{a}(g)=-1\\}.$ (2.10)

Then $G_{0}$ is a subgroup of $G$ of order $n/2$ and $G_{1}$ is a coset of
$G_{0}$. By (2.7), we have

$e^{\imath t\lambda_{g}}=\left\\{\begin{array}[]{cc}\alpha+\beta,&\mbox{ if
$g\in G_{0}$},\\\ \alpha-\beta,&\mbox{ if $g\in G_{1}$}.\end{array}\right.$
(2.11)

Assume the polar decompositions of $\alpha$ and $\beta$ are respective
$r_{\alpha}e^{\imath\theta_{\alpha}}$ and $r_{\beta}e^{\imath\theta_{\beta}}$.
According to $|\alpha|^{2}+|\beta|^{2}=1$ and
$\overline{\alpha}\beta+\alpha\overline{\beta}=0$, we have

$\left\\{\begin{array}[]{ll}r_{\alpha}^{2}+r_{\beta}^{2}&=1\\\
\theta_{\alpha}-\theta_{\beta}&=\frac{\pi}{2}+k\pi,\mbox{ for some
$k\in\mathbb{Z}$}.\end{array}\right.$ (2.12)

Thus

$\alpha\pm\beta=e^{\imath\theta_{\beta}}((-1)^{k}r_{\alpha}\imath\pm
r_{\beta}),$

Write $r_{\beta}+(-1)^{k}r_{\alpha}\imath=e^{\imath\theta_{0}}$. Then
$\theta_{0}\in\mathbb{R}$ and

$\alpha+\beta=e^{\imath(\theta_{0}+\theta_{\beta})},\alpha-\beta=e^{\imath({\pi}-\theta_{0}+\theta_{\beta})}.$
(2.13)

Write $t=2\pi T$ and $\theta_{0}=2\pi u_{0},\theta_{\beta}=2\pi u_{\beta}$.
Due to (2.11), it holds that

$T\lambda_{g}-u_{0}-u_{\beta}\in\mathbb{Z},\forall g\in G_{0},\mbox{ and
}T\lambda_{g}+u_{0}-u_{\beta}\in\frac{1}{2}+\mathbb{Z},\forall g\in G_{1}.$
(2.14)

Therefore,

$T\left(\sum_{g\in G_{0}}\lambda_{g}+\sum_{g\in
G_{1}}\lambda_{g}\right)-nu_{\beta}\in\frac{n}{4}+\mathbb{Z}.$

Since $\sum_{g\in G_{0}}\lambda_{g}+\sum_{g\in G_{1}}\lambda_{g}=\sum_{g\in
G}\lambda_{g}=0$, we have $u_{\beta}\in\mathbb{Q}$.

Now, we arrive at

$\sum_{g\in G_{0}}\lambda_{g}=\sum_{g\in G_{0}}\sum_{s\in
S}\chi_{g}(s)=\sum_{s\in S}\sum_{g\in G_{0}}\chi_{g}(s)=\sum_{s\in
S}\sum_{g\in
G}\frac{1+\chi_{a}(g)}{2}\chi_{g}(s)=\left\\{\begin{array}[]{cc}n/2,&\mbox{ if
$a\in S$,}\\\ 0,&\mbox{ otherwise}.\end{array}\right.$

Thus, $\sum_{g\in G_{0}}\lambda_{g}\in\mathbb{Z}$ and so $\sum_{g\in
G_{1}}\lambda_{g}\in\mathbb{Z}$ as well.

Let $d=|S|$. Thanks to (2.14), we deduce that

$T\left(n/2\cdot d+\sum_{g\in G_{1}}\lambda_{g}\right)-n/2\cdot u_{\beta}\in
n/4+\mathbb{Z},$

Thus we obtain that $T\in\mathbb{Q}$. By (2.14) again, we have $Td-
u_{0}-u_{\beta}\in\mathbb{Z}$ which further implies that $u_{0}\in\mathbb{Q}$,
and finally, we get that $\lambda_{g}\in\mathbb{Q}$ for every $g\in G$. Since
$\lambda_{g}$ is an algebraic integer, we know that
$\lambda_{g}\in\mathbb{Z}$. In other words, $\Gamma$ is an integral graph.
Meanwhile, since $T$ is rational, $e^{\imath t}$ is a root of unity.
Therefore, we have the following result.

######  Theorem 2.2.

Let $G$ be an abelian group of an even order $n$ and let $S$ be a symmetric
subset of $G$. Let $\Gamma={\rm Cay}(G,S)$ be a Cayley graph with the
adjacency matrix $A$. Suppose that $\alpha$ and $\beta$ are two nonzero
complex numbers. Then $\Gamma$ has $(\alpha,\beta)$-FR from a vertex $x$ to a
vertex $y$ at some time $t$ if and only if the following conditions hold.

(1) $a=x-y$ is of order two;

(2) $\Gamma$ is an integral graph;

(3) for every $g\in G_{0}$, $e^{\imath t\lambda_{g}}=\alpha+\beta$; for every
$g\in G_{1}$, $e^{\imath t\lambda_{g}}=\alpha-\beta$, where $G_{0}$ and
$G_{1}$ are defined by (2.10).

###### Proof.

The necessity has been proved by the foregoing discussion. Next, we prove the
sufficiency. Since for every $g\in G_{0}$, $e^{\imath
t\lambda_{g}}=\alpha+\beta$, and for every $g\in G_{1}$, $e^{\imath
t\lambda_{g}}=\alpha-\beta$, by (2.3), we get that, for every $x\in G$,

$H(t)_{x,x}=\frac{1}{n}\sum_{z\in G}e^{\imath
t\lambda_{z}}=\frac{1}{n}\left[\sum_{z\in G_{0}}(\alpha+\beta)+\sum_{z\in
G_{1}}(\alpha-\beta)\right]=\alpha.$

Meanwhile,

$\displaystyle H(t)_{x,x+a}$ $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{z\in G}e^{\imath t\lambda_{z}}\chi_{z}(a)$
$\displaystyle=$ $\displaystyle\frac{1}{n}\left[\sum_{z\in
G_{0}}(\alpha+\beta)\chi_{z}(a)+\sum_{z\in
G_{1}}(\alpha-\beta)\chi_{z}(a)\right]$ $\displaystyle=$
$\displaystyle\frac{\alpha}{n}\sum_{z\in
G}\chi_{z}(a)+\frac{\beta}{n}\left[\sum_{z\in G_{0}}\chi_{z}(a)-\sum_{z\in
G_{1}}\chi_{z}(a)\right].$

Due to $a$ is of order two, and $\sum_{z\in G}\chi_{z}(a)=0$, we get

$H(t)_{x,x+a}=\frac{2\beta}{n}\sum_{z\in
G_{0}}\chi_{z}(a)=\frac{2\beta}{n}\sum_{z\in G_{0}}1=\beta.$

Moreover, for every $b\neq 0,a$, we have

$\displaystyle H(t)_{x,x+b}$ $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{z\in G}e^{\imath t\lambda_{z}}\chi_{z}(b)$
$\displaystyle=$ $\displaystyle\frac{1}{n}\left[\sum_{z\in
G_{0}}(\alpha+\beta)\chi_{z}(b)+\sum_{z\in
G_{1}}(\alpha-\beta)\chi_{z}(b)\right]$ $\displaystyle=$
$\displaystyle\frac{\alpha}{n}\sum_{z\in
G}\chi_{z}(b)+\frac{\beta}{n}\left[\sum_{z\in G_{0}}\chi_{z}(b)-\sum_{z\in
G_{1}}\chi_{z}(b)\right]$ $\displaystyle=$
$\displaystyle\frac{2\beta}{n}\sum_{z\in G_{0}}\chi_{z}(b)$ $\displaystyle=$
$\displaystyle\frac{2\beta}{n}\sum_{z\in G}\frac{1+\chi_{z}(a)}{2}\chi_{z}(b)$
$\displaystyle=$ $\displaystyle\frac{\beta}{n}\sum_{z\in G}\chi_{z}(a+b)$
$\displaystyle=$ $\displaystyle 0.$

Therefore, we have $|\alpha|^{2}+|\beta|^{2}=1$, and $H(t)=\alpha I_{n}+\beta
Q$, where $Q=\sum_{g\in G}e_{g}e_{g+a}^{T}$ is a symmetric permutation matrix.
By Lemma 2.1, $\Gamma$ has $(\alpha,\beta)$-FR at time $t$. ∎

######  Remark 1.

According to Definition 1.1, the existence of $(\alpha,\beta)$-FR is
equivalent to the existence of $e^{\imath\theta}(\alpha,\beta)$-FR with
$\theta,\alpha$ being real numbers and $\beta$ being a complex number. Based
on Theorem 2.2, we claim that $\theta$ is a rational number and $\beta$ is a
pure complex number if $\Gamma={\rm Cay}(G,S)$ has
$e^{\imath\theta}(\alpha,\beta)$-FR. Due to Lemma 2.1, it is easy to check
that $\beta$ is a pure complex number. By (2.11), we obtain that

$e^{\imath
t\lambda_{g}-\imath\theta}=\left\\{\begin{array}[]{cc}\alpha+\beta,&\mbox{ if
$g\in G_{0}$},\\\ \alpha-\beta,&\mbox{ if $g\in G_{1}$}.\end{array}\right.$

Since $\alpha$ is a real number and $\beta$ is a pure complex number, we
deduce that $e^{\imath t(|S|+\lambda_{g})-2\imath\theta}=1$ with $g\in G_{1}$.
Thanks to Items (2) and (3) in Theorem 2.2, $\theta$ is a rational number.

In what follows, we continue to investigate the conditions in Theorem 2.2.

Taking an fixed element $g_{1}$ in $G_{1}$, by (2.11), we have

$T(d-\lambda_{g})\in\mathbb{Z}\mbox{ for every $g\in G_{0}$ and
}T(\lambda_{g}-\lambda_{g_{1}})\in\mathbb{Z}\mbox{ for every $g\in G_{1}$}.$
(2.15)

Since $\Gamma$ is an integral graph, we define the following integers:

$M_{0}=\gcd(d-\lambda_{g}:g\in
G_{0}),M_{1}=\gcd(\lambda_{g_{1}}-\lambda_{g}:g\in
G_{1}),M=\gcd(M_{0},M_{1}).$ (2.16)

(2.15) is equivalent to

$TM_{0},TM_{1}\in\mathbb{Z},$

and thus

$T\in\frac{1}{M}\mathbb{Z}.$ (2.17)

Next, we show that $M$ is a divisor of $n=|G|$.

######  Lemma 2.3.

Let $G$ be an abelian group of order $n$ and let ${\rm Cay}(G,S)$ be an
integral graph. Then $M$ is a divisor of $n$, where $M$ is defined by (2.16).

###### Proof.

Write

$d=\lambda_{g}+Mt_{g}\mbox{ for every $g\in G_{0}$ and
$\lambda_{g}=\lambda_{g_{1}}+Ms_{g}$ for every $g\in G_{1}$, where
$t_{g},s_{g}\in\mathbb{Z}$}.$

Take an element $z\in S$ with $z\neq 0,a$. Then

$\sum_{g\in G}\lambda_{g}\overline{\chi_{g}(z)}=\sum_{s\in S}\sum_{g\in
G}\chi_{g}(s-z)=n.$ (2.18)

Meanwhile,

$\displaystyle\sum_{g\in G}\lambda_{g}\overline{\chi_{g}(z)}$ $\displaystyle=$
$\displaystyle\sum_{g\in G_{0}}(d-t_{g}M)\overline{\chi_{g}(z)}+\sum_{g\in
G_{0}}(\lambda_{g_{1}}-s_{g}M)\overline{\chi_{g}(z)}$ (2.19) $\displaystyle=$
$\displaystyle(d+\lambda_{g_{1}})\sum_{g\in
G_{0}}\overline{\chi_{g}(z)}-M\sum_{g\in
G_{1}}(t_{g}+s_{g})\overline{\chi_{g}(z)}.$

Moreover,

$\sum_{g\in G_{0}}\overline{\chi_{g}(z)}=\sum_{g\in
G}\frac{1+\chi_{g}(a)}{2}\overline{\chi_{g}(z)}=\sum_{g\in G}\chi_{g}(a-z)=0.$
(2.20)

(2.18)-(2.20) imply that

$n/M=-\sum_{g\in G_{1}}(t_{g}+s_{g})\overline{\chi_{g}(z)}.$ (2.21)

The LHS of (2.21) is in $\mathbb{Q}$, while the RHS is an algebraic integer.
Thus $n/M\in\mathbb{Z}$. ∎

By Lemma 2.3, we have the following result.

######  Corollary 2.4.

Let $G$ be an abelian group of order $n$ and let ${\rm Cay}(G,S)$ be an
integral graph. If ${\rm Cay}(G,S)$ has FR at some time $t$, then $e^{\imath
t}$ is an $n$-th root of unity.

###### Proof.

It is a direct consequence of (2.17) and Lemma 2.3. ∎

According to (2.17), we show the non-existence of FR on integral abelian
Cayley graphs if $M=1$.

######  Proposition 2.5.

Let $\Gamma={\rm Cay}(G,S)$ be an integral abelian Cayley graph. Let $M$ be
defined by in (2.16). If $M=1$, then $\Gamma$ cannot provide FR between any
two vertices.

###### Proof.

If $M=1$, then by (2.17), we know that $T\in\mathbb{Z}$ which means that
$t=2\pi$. The desired result follows from the fact $H(t)=I_{n}$.∎

Thanks to Theorem 2.2, if one wants to construct graphs admitting FR between
two distinct vertices, the feasible graphs are integral and the difference of
two vertices has even order. Using this fact, we propose a general
construction of abelian Cayley graphs having FR, as the following corollary
reveals. The proof of the following corollary is based on a direct
computation. Of course, one can use Theorem 2.2 to get another proof of this
result as well.

######  Corollary 2.6.

Let $G=\mathbb{Z}_{2}^{s}\times H$ such that $s\geq 1$ and $H$ is an abelian
group of order $\ell$. Let $\Gamma={\rm Cay}(G,S)$ be an integral graph and
let $M$ be defined in (2.16). If $\frac{|S|}{M}\notin\frac{1}{4}\mathbb{Z}$,
then $\Gamma={\rm Cay}(G,S)$ has
$\left(\cos\left(\frac{2\pi|S|}{M}\right),\imath\sin\left(\frac{2\pi|S|}{M}\right)\right)$-FR
between $(x,y)$ and $(x+a,y)$ for every $(x,y)\in G$ and $0\neq
a\in\mathbb{Z}_{2}^{s}$ at time $t=\frac{2\pi}{M}$.

###### Proof.

Denote vertices $(x,y)$ and $(x+a,y)$ by respective $\mu$ and $\nu$. It is
well known that $\mathbb{Z}_{2}^{s}$ is isomorphic to the additive group of
${\mathbb{F}}_{2^{s}}$, the finite field with $2^{s}$ elements. Given
$g\in{\mathbb{F}}_{2^{s}}$, an additive character of ${\mathbb{F}}_{2^{s}}$ is
given by $\chi_{g}(h)=(-1)^{\operatorname{Tr}^{s}_{1}(gh)}$ with
$h\in{\mathbb{F}}_{2^{s}}$ and $\operatorname{Tr}^{s}_{1}(\cdot)$ being the
trace function mapping from ${\mathbb{F}}_{2^{s}}$ to ${\mathbb{F}}_{2}$. We
refer readers to [27, Chapter 5] for more details on additive characters.
Below, we view $G$ as $G=\mathbb{F}_{2^{s}}\times H$.

By (2.3), we obtain that

$H(t)_{\mu,\mu}=\frac{1}{2^{s}\ell}\sum_{z\in G}e^{\imath t\lambda_{z}}\ \
\hbox{and}\ \ H(t)_{\mu,\nu}=\frac{1}{2^{s}\ell}\sum_{z=(z_{1},z_{2})\in
G}e^{\imath t\lambda_{z}}(-1)^{\operatorname{Tr}^{s}_{1}(az_{1})}.$

Thanks to
$|\\{t\in{\mathbb{F}}_{2^{s}}:\chi_{c}(t)=0\\}|=|\\{t\in{\mathbb{F}}_{2^{s}}:\chi_{c}(t)=1\\}|=2^{s-1}$,
we deduce that

$\displaystyle\left|H(2\pi/M)_{\mu,\mu}\right|^{2}+\left|H(2\pi/M)_{\mu,\nu}\right|^{2}$
$\displaystyle=$ $\displaystyle\frac{1}{2^{2s}\ell^{2}}\sum_{z\in G}\sum_{g\in
G}e^{\imath 2\pi\frac{\lambda_{z}-\lambda_{g}}{M}}$
$\displaystyle+\frac{1}{2^{2s}\ell^{2}}\sum_{z=(z_{1},z_{2})\in
G}\sum_{g=(g_{1},g_{2})\in G}e^{\imath
2\pi\frac{\lambda_{z}-\lambda_{g}}{M}}(-1)^{\operatorname{Tr}^{s}_{1}(az_{1}+ag_{1})}$
$\displaystyle=$
$\displaystyle\frac{1}{2^{2s}\ell^{2}}\sum_{z=(z_{1},z_{2})\in
G}\sum_{g=(g_{1},g_{2})\in
G}\left(1+(-1)^{\operatorname{Tr}^{s}_{1}(az_{1}+ag_{1})}\right)$
$\displaystyle=$ $\displaystyle 1.$

By (2.7) and (2.11), there exist a rational number $v$, a real number $\alpha$
and a pure complex number $\beta$ such that
$e^{\imath\frac{2\pi}{M}|S|}=e^{\imath 2\pi v}(\alpha+\beta)$. It can be
easily seen that $\alpha$ and $\beta$ are nonzero if
$\frac{|S|}{M}+v\notin\frac{1}{4}\mathbb{Z}$. Putting $v=0$, it follows from
$\frac{|S|}{M}\notin\frac{1}{4}\mathbb{Z}$ that $\Gamma={\rm Cay}(G,S)$ has
$\left(\cos\left(\frac{2\pi|S|}{M}\right),\imath\sin\left(\frac{2\pi|S|}{M}\right)\right)$-FR
at time $t=\frac{2\pi}{M}$. ∎

######  Remark 2.

(1)By the proof of the above corollary, if there exist a rational number $v$,
a real number $\alpha$ and a pure complex number $\beta$ such that
$e^{\imath\frac{2\pi}{M}|S|}=e^{\imath 2\pi v}(\alpha+\beta)$ and
$\frac{|S|}{M}+v\notin\frac{1}{4}\mathbb{Z}$, then the graph $\Gamma={\rm
Cay}(G,S)$ defined in Corollary 2.6 has $e^{\imath 2\pi v}(\alpha,\beta)$-FR
between $(x,y)$ and $(x+a,y)$ for every $(x,y)\in G$ and $0\neq
a\in\mathbb{Z}_{2}^{s}$ at time $t=\frac{2\pi}{M}$. For simplicity, we state
the case $v=0$ in Corollary 2.6.

(2)Since we only discuss the case $\alpha\beta\neq 0$, the condition
$\frac{|S|}{M}\notin\frac{1}{4}\mathbb{Z}$ (resp.
$\frac{|S|}{M}+v\notin\frac{1}{4}\mathbb{Z}$) is needed in Corollart 2.6.
Removing the condition $\frac{|S|}{M}\notin\frac{1}{4}\mathbb{Z}$ (resp.
$\frac{|S|}{M}+v\notin\frac{1}{4}\mathbb{Z}$), the graph $\Gamma={\rm
Cay}(G,S)$ in Corollary 2.6 is said to have PST, FR or be periodic.

Using Corollary 2.6, we construct the following family of abelian Cayley
graphs having FR.

######  Proposition 2.7.

Let $p$ be a prime and let $r\geq 1$ be an integer. Let
$G=\mathbb{Z}_{2}\times\mathbb{Z}_{p^{r}}\times H$ with $H$ being an abelian
group of order $m$. Let $S$ be a symmetric subset in $G$ which is defined by

$S=\\{(0,\ell,h):1\leq\ell\leq p^{r}-1,\gcd(\ell,p)=1,h\in
H\\}\cup\\{(1,0,0)\\}.$

If $p^{r-1}m\neq 1,2,4$, then $\Gamma={\rm Cay}(G,S)$ has FR between $(x,y,z)$
and $(x+1,y,z)$ for every $(x,y,z)\in G$ at the time
$t=\frac{2\pi}{p^{r-1}m}$.

###### Proof.

It is easy to see that the dual group of $G$ is

$\widehat{G}=\\{\varphi_{u}\chi_{v}\psi_{w}:u\in\mathbb{Z}_{2},v\in\mathbb{Z}_{p^{r}},w\in
H\\}$

where
$\varphi_{u}\chi_{v}\psi_{w}(x,y,z)=(-1)^{ux}\omega_{p^{r}}^{vy}\psi_{w}(z)$,
and $\omega_{v}=e^{\imath\frac{2\pi}{v}}$ is a primitive $v$-th root of unity.
Therefore, the eigenvalues of $\Gamma$ are

$\displaystyle\lambda_{x,y,z}$ $\displaystyle=$ $\displaystyle\sum_{(u,v,w)\in
S}\varphi_{u}\chi_{v}\psi_{w}(x,y,z)$ $\displaystyle=$
$\displaystyle\sum_{1\leq\ell\leq p^{r}-1,\gcd(\ell,p)=1}\omega_{p^{r}}^{\ell
y}\sum_{w\in H}\psi_{w}(z)+(-1)^{x},\ \ (x,y,z)\in G.$

The summation

$\sum_{1\leq\ell\leq p^{r}-1,\gcd(\ell,p)=1}\omega_{p^{r}}^{\ell
y}:=c(y,p^{r})$

is in fact a Ramanujan function [28, Corollary 2.4] and

$c(y,p^{r})=\left\\{\begin{array}[]{ll}-p^{r-1},&\mbox{ if
$v_{p}(y)={r-1}$},\\\ p^{r-1}(p-1)&\mbox{ if $y=0$},\\\ 0,&\mbox{
otherwise}.\end{array}\right.$

Here we identify the elements in $\mathbb{Z}_{p^{r}}$ as integers and
$v_{p}(y)$ is the highest power of $p$ that divides $y$. Then we derive that

$\lambda_{x,y,z}=\left\\{\begin{array}[]{ll}(-1)^{x}+(-p^{r-1})m,&\mbox{ if
$z=0$ and $v_{p}(y)={r-1}$},\\\ (-1)^{x}+p^{r-1}(p-1)m,&\mbox{ if $z=0$ and
$y=0$},\\\ (-1)^{x},&\mbox{ otherwise.}\end{array}\right.$

Since $a=(x+1,y,z)-(x,y,z)=(1,0,0)$, it is obvious that

$G_{0}=\\{(0,y,h):y\in\mathbb{Z}_{p^{r}},h\in H\\},\mbox{ and
}G_{1}=\\{(1,y,h):y\in\mathbb{Z}_{p^{r}},h\in H\\}.$

We have $M_{0}=M_{1}=M=p^{r-1}m$, which are defined by (2.16). Due to
$\frac{|S|}{M}=p-1+\frac{1}{p^{r-1}m}\notin\frac{1}{4}\mathbb{Z}$ and
Corollary 2.6 or Theorem 2.2 directly, $\Gamma$ has
$\left(\cos\left(\frac{2\pi}{p^{r-1}m}\right),\imath\sin\left(\frac{2\pi}{p^{r-1}m}\right)\right)$-FR
between $(x,y,z)$ and $(x+1,y,z)$ for every $(x,y,z)\in G$ at the time
$t=\frac{2\pi}{p^{r-1}m}$. ∎

The following example is used to illustrate Proposition 2.7.

######  Example 2.8.

Let $G=\mathbb{Z}_{2}\times\mathbb{Z}_{9}$ and
$S=\\{(0,1),(0,2),(0,4),(0,5),(0,7),(0,8),(1,0)\\}$. Then $\Gamma={\rm
Cay}(G,S)$ has FR between $(0,0)$ and $(1,0)$ at time $t=2\pi/3$.

###### Proof.

The eigenvalues of $\Gamma$ are as follows.

$\lambda_{x,y}=\omega_{9}^{y}+\omega_{9}^{2y}+\omega_{9}^{4y}+\omega_{9}^{5y}+\omega_{9}^{7y}+\omega_{9}^{8y}+(-1)^{x},(x,y)\in
G.$

Thus,

$\lambda_{0,0}=7,\lambda_{0,3}=\lambda_{0,6}=-2,\lambda_{0,y}=1,y\in\mathbb{Z}_{9}^{*}:=\\{1,2,4,5,7,8\\},\mbox{
and}$
$\lambda_{1,0}=5,\lambda_{1,3}=\lambda_{1,6}=-4,\lambda_{1,y}=-1,y\in\mathbb{Z}_{9}^{*}.$

It is obvious that

$G_{0}=\\{(0,y):y\in\mathbb{Z}_{9}\\},G_{0}=\\{(1,y):y\in\mathbb{Z}_{9}\\}.$

Thus, $M_{0}=M_{1}=M=3$. Moreover,

$e^{\imath\frac{2\pi}{3}\lambda_{g}}=\left\\{\begin{array}[]{cc}e^{\imath\frac{2\pi}{3}},&g\in
G_{0},\\\ e^{-\imath\frac{2\pi}{3}},&g\in G_{1}.\end{array}\right.$

Thus $\Gamma$ has $(\cos(2\pi/3),\imath\sin(2\pi/3))$-FR at time $t=2\pi/3$.

Indeed, by a Magma program, we find that

$H(2\pi/3)={\rm diag}\left(\underset{9\ \
cpoies}{\underbrace{{\left(\begin{array}[]{cc}-\frac{1}{2}&\frac{\sqrt{3}}{2}\\\
\frac{\sqrt{3}}{2}&-\frac{1}{2}\\\
\end{array}\right),\cdots,\left(\begin{array}[]{cc}-\frac{1}{2}&\frac{\sqrt{3}}{2}\\\
\frac{\sqrt{3}}{2}&-\frac{1}{2}\\\ \end{array}\right)}}}\right).$

Thus $\Gamma$ has $(-\frac{1}{2},\frac{\sqrt{3}}{2})$-FR at time $t=2\pi/3$. ∎

In Proposition 2.7, if $p^{r-1}m=1,2\ \mbox{or}\ 4$, the Cayley graph
$\Gamma={\rm Cay}(G,S)$ may have FR. We do not have a general result like
Corollary 2.7 to guarantee the existence of FR on $\Gamma$ whenever
$p^{r-1}m=1,2\ \mbox{or}\ 4$. Using Theorem 2.2, we can check the existence of
FR on $\Gamma$ under these cases one by one. The following example shows how
to verify it when $p=3$, $r=1$ and $m=1$.

######  Example 2.9.

Let $G=\mathbb{Z}_{2}\times\mathbb{Z}_{3}$ and $S=\\{(0,1),(0,2),(1,0)\\}$.
Then the Cayley graph $\Gamma={\rm Cay}(G,S)$ has FR at time $t=2\pi/3$.

###### Proof.

One can check that the spectra of $\Gamma$ are

$\lambda_{0,0}=3,\lambda_{0,1}=\lambda_{0,2}=0,\lambda_{1,0}=1,\lambda_{1,1}=\lambda_{1,2}=-2.$

Meanwhile,

$G_{0}=\\{(0,0),(0,1),(0,2)\\},G_{1}=\\{(1,0),(1,1),(1,2)\\}.$

It is easy to check that $M=3$ and

$\displaystyle e^{\imath\frac{2\pi}{3}\lambda_{g}}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cc}1,&g\in G_{0},\\\
\cos(2\pi/3)+\imath\sin(2\pi/3),&g\in G_{1}.\end{array}\right.$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cc}e^{\imath\frac{\pi}{3}}(\frac{1}{2}-\imath\frac{\sqrt{3}}{2}),&g\in
G_{0},\\\ e^{\imath\frac{\pi}{3}}(\frac{1}{2}+\imath\frac{\sqrt{3}}{2}),&g\in
G_{1}.\end{array}\right.$

By Theorem 2.2, $\Gamma$ has
$e^{\imath\frac{\pi}{3}}\left(\frac{1}{2},-\imath\frac{\sqrt{3}}{2}\right)$ at
time $t=2\pi/3$. ∎

In Corollary 2.6, we propose a general construction of Cayley graphs over
$G=\mathbb{Z}_{2}^{s}\times H$ having FR. Next, we discuss the existence of FR
on Cayley graphs over $G=\mathbb{Z}_{2^{s}}\times H$.

######  Corollary 2.10.

Let $G=\mathbb{Z}_{2^{s}}\times H$ such that $s\geq 1$ and $H$ is an abelian
group of order $\ell$. Let $\Gamma={\rm Cay}(G,S)$ be an integral graph and
let $M$ be defined in (2.16). If $\frac{|S|}{M}\notin\frac{1}{4}\mathbb{Z}$,
then $\Gamma={\rm Cay}(G,S)$ has
$\left(\cos\left(\frac{2\pi|S|}{M}\right),\imath\sin\left(\frac{2\pi|S|}{M}\right)\right)$-FR
between $(x,y)$ and $(x+2^{s-1},y)$ for every $(x,y)\in G$ at time
$t=\frac{2\pi}{M}$.

###### Proof.

The proof of this corollary can be completed by the method analogous to that
used in Corollary 2.6 ∎

At the end of this section, we propose another family of Cayley graphs
admitting FR.

######  Proposition 2.11.

Let $r_{0},r_{1},\cdots,r_{s}$ be positive integers and let
$p_{0}=2,p_{1},\cdots,p_{s}$ be distinct primes such that
$\prod_{j=0}^{s}p_{j}^{r_{j}-1}\neq 1,2,4$ with $s\geq 1$. Let
$G=\prod_{j=0}^{s}\mathbb{Z}_{p_{j}^{r_{j}}}$ and let $S$ be a subset of $G$
defined by

$S=\\{(\ell_{0},\ell_{1},\cdots,\ell_{s}):1\leq\ell_{i}\leq
p_{i}^{r_{i}}-1,\gcd(\ell_{i},p_{i})=1,0\leq i\leq
s\\}\cup\\{(2^{r_{0}-1},0,\cdots,0)\\}.$

Let $\Gamma={\rm Cay}(G,S)$ be a Cayley graph. Then $\Gamma$ has FR between
$(x_{0},x_{1},\cdots,x_{s})$ and $(x_{0}+2^{r_{0}-1},x_{1},\cdots,x_{s})$ for
every $(x_{0},x_{1},\cdots,x_{s})\in G$ at time
$2\pi/\prod_{i=0}^{s}p_{i}^{r_{i}-1}$.

###### Proof.

Due to
$a=(x_{0}+2^{r_{0}-1},x_{1},\cdots,x_{s})-(x_{0},x_{1},\cdots,x_{s})=(2^{r_{0}-1},0,\cdots,0)$,
it is easy to see that

$G_{0}=\\{(2x,y_{1},\cdots,y_{s}):x\in\mathbb{Z}_{2^{r_{0}}},y_{i}\in\mathbb{Z}_{p_{i}^{r_{i}}}\\},G_{1}=\\{(2x+1,y_{1},\cdots,y_{s}):x\in\mathbb{Z}_{2^{r_{0}}},y_{i}\in\mathbb{Z}_{p_{i}^{r_{i}}}\\}.$

For $\mathbf{x}=(x_{0},x_{1},\cdots,x_{s})$, the corresponding eigenvalue of
$\Gamma$ is

$\lambda_{\mathbf{x}}=(-1)^{x_{0}}+\sum_{\underset{1\leq\ell_{j}\leq
p_{j}^{r_{j}}-1,\gcd(p_{j},\ell_{j})=1}{\ell_{0},\ell_{1},\cdots,\ell_{s}}}\prod_{j=0}^{s}\xi_{{p_{j}}^{r_{j}}}^{\ell_{j}x_{j}}.$

Thus,

$\lambda_{\mathbf{x}}=\left\\{\begin{array}[]{ll}1+\prod_{i=0}^{s}p_{i}^{r_{1}-1}(p_{1}-1),&\mbox{
if $(x_{0},x_{1},\cdots,x_{s})=\mathbf{0}$},\\\ 1+A,&\mbox{ if $x_{j}\neq
0,v_{p_{j}}(x_{j})=r_{j}-1,$ for some $j\in I\subseteq\\{0,\cdots,s\\}$},\\\
(-1)^{x_{0}},&\mbox{ if there is some $x_{i}\neq
0,v_{p_{i}}(x_{i})<{r_{i}-1}$}.\end{array}\right.$

where $A=(-1)^{|I|}\prod_{j\in I}p_{j}^{r_{j}-1}\prod_{i\not\in
I}p_{i}^{r_{i}-1}(p_{i}-1)$. It follows that
$M=\prod_{i=0}^{s}p_{i}^{r_{i}-1}$, where $M$ is defined by (2.16). Due to
$|S|/M\notin\frac{1}{4}\mathbb{Z}$ and Corollary 2.10, $\Gamma$ has
$\left(\cos(2\pi/M),\imath\sin(2\pi/M)\right)$-FR at time $2\pi/M$. ∎

######  Remark 3.

(1) It was shown in [10] that the order of the underlying group should be
doubly-even if an abelian Cayley graph has perfect state transfer. However,
for FR, Propositions 2.7 and 2.11 show that there are many more graphs having
FR than that for PST on abelian Cayley graphs.

(2) Proposition 2.11 says that for every even number $n$ having a square
divisor of an odd prime power, there is an abelian Cayley graph having FR with
the underlying group of order $n$.

##  3\. Plateaued functions over abelian groups and FR on abelian Cayley
graphs

Let $G$ be an abelian group of order $n$ and let $\mathbb{C}^{G}$ be the set
of complex-valued functions on $G$. Then $\mathbb{C}^{G}$ is an
$n$-dimensional vector space. Given $f,g\in\mathbb{C}^{G}$, define a scalar
product as

$\langle f,g\rangle=\sum_{x\in G}f(x)\overline{g(x)}.$

Then $\mathbb{C}^{G}$ is a Hilbert space with
$\\{\frac{1}{\sqrt{n}}\chi_{x}:x\in G\\}$ an orthnormal basis. Thus every
function $f$ has the following decomposition.

$f(x)=\sum_{z\in G}\langle
f,\frac{1}{\sqrt{n}}\chi_{z}\rangle\frac{1}{\sqrt{n}}\chi_{z}(x).$

For any $f\in\mathbb{C}^{G}$, its Fourier transform is defined by

$\widehat{f}(\chi_{z})=\sum_{x\in G}f(x)\overline{\chi_{z}(x)}=\langle
f,\chi_{z}\rangle.$

######  Definition 3.1.

[29] Let $G$ be an abelian group of order $n$. A function $f\in\mathbb{C}^{G}$
is called bent on $G$ if for every $z\in G$, we have

$|\widehat{f}(\chi_{z})|=\sqrt{n}.$

Bent functions over finite fields of even characteristic were invented and
named in 1976 by Rothaus [30]. As a generalization of bent functions, the so-
called plateaued functions over finite fields of characteristic two were first
introduced by Zheng and Zhang in [31].

######  Definition 3.2.

An $n$-variable Boolean function is said to be $r$-plateaued if the values of
its Walsh transform (the Fourier transform of $(-1)^{f}$) belong to the set
$\\{0,\pm 2^{\frac{n+r}{2}}\\}$ for some fixed $r$.

Bent and plateaued functions have many applications in communications and
cryptography, for a comprehensive survey on such functions, interested readers
may refer to [32].

Let $G$ be an abelian group of order $n$ and let $e=\exp(G)$ be the largest
order of the elements in $G$. Then the values of $\chi(z)(z\in
G,\chi\in\hat{G})$ are in the cyclotomic field $\mathbb{Q}(\omega_{e})$. The
Galois group of the extension $\mathbb{Q}(\omega_{e})/\mathbb{Q}$ is
isomorphic to

$\mathbb{Z}_{e}^{*}=\\{\ell:1\leq\ell\leq e-1,\gcd(\ell,e)=1\\}.$

For every $\ell\in\mathbb{Z}_{e}^{*}$, the corresponding isomorphic in ${\rm
Gal}(\mathbb{Q}(\omega_{e})/\mathbb{Q})$ is determined by

$\sigma_{\ell}:\omega_{e}\mapsto\omega_{e}^{\ell}.$

######  Definition 3.3.

A function $f\in\mathbb{C}^{G}$ is called a class function over $G$ if $f$ is
integer-valued, namely, $f(x)\in\mathbb{Z}$ for every $x\in G$, and for every
$\ell\in\mathbb{Z}_{e}^{*}$, it holds that

$f(\ell x)=f(x),\forall x\in G.$ (3.1)

######  Lemma 3.4.

Let $G$ be an abelian group of order $n$ and let $f\in\mathbb{C}^{G}$ be a
class function over $G$. Then for every $\chi\in\hat{G}$, we have

$\hat{f}(\chi)\in\mathbb{Z}.$

###### Proof.

Since $f$ is integer-valued,

$\hat{f}(\chi_{g})=\sum_{x\in G}f(x)\chi_{g}(x)$

is contained in the integer ring $\mathbb{Z}[\omega_{e}]$, and for every
$\ell\in\mathbb{Z}_{e}^{*}$,

$\displaystyle\sigma_{\ell}(\hat{f}(\chi_{g}))$ $\displaystyle=$
$\displaystyle\sum_{x\in G}f(x)\sigma_{\ell}(\chi_{g}(x))=\sum_{x\in
G}f(x)(\chi_{g}(\ell x))$ $\displaystyle=$ $\displaystyle\sum_{x\in G}f(\ell
x)(\chi_{g}(\ell x))\ \ (\mbox{ since $f$ is a class function})$
$\displaystyle=$ $\displaystyle\sum_{x\in G}f(x)\chi_{g}(x)$ $\displaystyle=$
$\displaystyle\hat{f}(\chi_{g}).$

Therefore,
$\hat{f}(\chi_{g})\in\mathbb{Q}\cap\mathbb{Z}[\omega_{e}]=\mathbb{Z}$. ∎

As an analogue to plateaued functions over finite fields of even
characteristic, we define plateaued functions over abelian groups as follows.

######  Definition 3.5.

Let $G$ be an abelian group of order $n$. Suppose that $p$ is a prime number.
Let $f\in\mathbb{C}^{G}$ be a class function such that for every
$\chi\in\widehat{G}$, the Fouerier transform of $f$ at $\chi$ satisfying

$\widehat{f}(\chi)\equiv k\pmod{p^{r}}$ (3.2)

for some fixed non-negative integers $k,r$. Then $f$ is called
$p^{r}$-plateaued over $G$, where $r$ is the greatest integer $r$ satisfying
(3.2).

It is easy to see that the classical plateaued functions over finite fields
with even characteristic are special cases of our plateaued functions over
abelian groups.

######  Example 3.6.

Let $G=\mathbb{Z}_{3}\times\mathbb{Z}_{3}$ and $S$ be a subset of $G$ defined
by

$S=\\{(1,0),(2,0),(0,1),(0,2)\\}.$

Let $f$ be the characteristic function of $S$, namely, $f(x)=1$ if $x\in S$
and $0$, otherwise. Then $f$ is a $3$-plateaued function over $G$.

###### Proof.

Indeed, since for every $\ell\in\mathbb{Z}_{3}^{*}=\\{1,2\\}$, we have $\ell
S:=\\{\ell x:x\in S\\}=S$. Thus $f$ is a class function. Moreover, for every
$g=(g_{1},g_{2})\in G$, we have

$\hat{f}(\chi_{g})=\sum_{x\in
G}f(x)\chi_{g}(x)=\omega_{3}^{g_{1}}+\omega_{3}^{-g_{1}}+\omega_{3}^{g_{2}}+\omega_{3}^{-g_{2}}.$

Hence, $\hat{f}(\chi_{g})\in\\{4,-2,1\\}$ and then $\hat{f}(\chi_{g})\equiv
1\pmod{3}$. So that $f$ is a $3$-plateaued function over $G$. ∎

Our main result in this section is as follows.

######  Theorem 3.7.

Let $H$ be an abelian group of order $n$ and $S_{1}$ be a subset of $H$
satisfying $0\notin S_{1}$ and $\ell S_{1}=S_{1}$ for all
$\ell\in\mathbb{Z}_{e}^{*}$. Let $G=\mathbb{Z}_{2}\times H$ and let $S$ be a
subset of $G$ defined by

$S=(1,S_{1})\cup(0,S_{1})\cup\\{(1,0)\\}.$

Assume that $p$ divides $d=|S_{1}|$. Let $f$ be the characteristic function of
$S_{1}$ and suppose that $f$ is an $p^{r}$-plateaued function over $H$ with
$r\geq 1$. Then the Cayley graph $\Gamma={\rm Cay}(G,S)$ has FR at time
$t=\pi/p^{r_{0}}$ between $(x+1,y)$ and $(x,y)$ for every $(x,y)\in G$, where
$r_{0}=\min(r,v_{p}(d))\geq 1$.

###### Proof.

For every $(x,y)\in G$, the corresponding eigenvalue of $\Gamma$ is

$\lambda_{x,y}=(-1)^{x}+(-1)^{x}\sum_{z\in S_{1}}\chi_{y}(z)+\sum_{z\in
S_{1}}\chi_{y}(z)=(-1)^{x}+((-1)^{x}+1)\hat{f}(\chi_{y}).$

Due to $a=(x+1,y)-(x,y)=(1,0)$, it is easy to see that

$G_{0}=\\{(0,h):h\in H\\},\mbox{ and }G_{1}=\\{(1,h):h\in H\\}.$

Thus

$\lambda_{x,y}=\left\\{\begin{array}[]{cc}1+2\hat{f}(\chi_{y}),&\mbox{ if
$(x,y)\in G_{0}$},\\\ -1,&\mbox{ if $(x,y)\in G_{1}$}.\end{array}\right.$

Therefore, for every $(0,y)\in G_{0}$, we have

$e^{\imath
t\lambda_{0,y}}=e^{\imath\frac{\pi}{p^{r_{0}}}(\lambda_{0,y}-\lambda_{0,0}+\lambda_{0,0})}=e^{\imath\frac{\pi}{p^{r_{0}}}(2(\hat{f}(\chi_{y})-\hat{f}(\chi_{0}))+1+2d)}.$

Since $\hat{f}(\chi_{y})-\hat{f}(\chi_{0})\equiv 0\pmod{p^{r}}$, and $d\equiv
0\pmod{p^{v_{p}(d)}}$, we get

$e^{\imath t\lambda_{0,y}}=e^{\imath\frac{\pi}{p^{r_{0}}}}.$

Meanwhile, we have

$e^{\imath t\lambda_{1,y}}=e^{-\imath\frac{\pi}{p^{r_{0}}}}.$

Thus we get the desired result by Theorem 2.2. ∎

Next we present an example to illustrate Theorem 3.7.

######  Example 3.8.

Let $H=\mathbb{Z}_{9}$ and let $S_{1}$ be defined by

$S_{1}=\\{1,2,4,5,7,8\\}.$

Let $G=\mathbb{Z}_{2}\times\mathbb{Z}_{9}$ and let $S$ be defined by

$S=\\{(0,1),(0,2),(0,4),(0,5),(0,7),(0,8),(1,1),(1,2),(1,4),(1,5),(1,7),(1,8),(1,0)\\}.$

Then $\Gamma={\rm Cay}(G,S)$ has FR at time $t=\pi/3$.

###### Proof.

Let $f$ be the characteristic function of the set $S_{1}$. Then for every
$y\in H$, we have

$\hat{f}(\chi_{y})=\sum_{x\in
H}f(x)\chi_{x}(y)=\omega_{9}^{y}+\omega_{9}^{2y}+\omega_{9}^{4y}+\omega_{9}^{5y}+\omega_{9}^{7y}+\omega_{9}^{8y}=c(y,9),$

where $c(y,9)$ is a Ramanujan function. Thus

$\widehat{f}(\chi_{0})=6,\widehat{f}(\chi_{1})=\widehat{f}(\chi_{2})=\widehat{f}(\chi_{4})=\widehat{f}(\chi_{5})=\widehat{f}(\chi_{7})=\widehat{f}(\chi_{8})=0,\widehat{f}(\chi_{3})=\widehat{f}(\chi_{6})=-3.$

Therefore, $f$ is a $3$-plateaued function. Moreover $3|d$. By Theorem 3.7,
$\Gamma$ has FR at time $\pi/3$.

We have checked the results by a Magma programme, it shows that

$H(\pi/3)={\rm diag}\left(\underset{9\ \
copies}{\underbrace{{\left(\begin{array}[]{cc}-\frac{1}{2}&\frac{\sqrt{3}}{2}\\\
\frac{\sqrt{3}}{2}&-\frac{1}{2}\\\
\end{array}\right),\cdots,\left(\begin{array}[]{cc}-\frac{1}{2}&\frac{\sqrt{3}}{2}\\\
\frac{\sqrt{3}}{2}&-\frac{1}{2}\\\ \end{array}\right)}}}\right).$

This completes the proof. ∎

Note that the transfer matrix at time $t=\pi/3$ in the above example is
exactly the same as that in Example 2.8. Combining these two examples, we know
that for a given abelian group of even order, there may have different choices
of the connection sets such that the resulting graphs have FR at a different
time.

##  4\. Cubelike graphs

Let $\mathbb{F}_{2^{n}}$ be the finite field with $2^{n}$ elements. A Cayley
graph ${\rm Cay}(G,S)$ is called a cubelike graph if $G$ is the additive group
of some finite field $\mathbb{F}_{2^{n}}$. Since $(\mathbb{F}_{2^{n}},+)$ is
an elementary $2$-group, the classical Hamming graph is a special kind of
cubelike graphs. Note that every nonzero element in $\mathbb{F}_{2^{n}}$ has
order two. Moreover, the map $x\mapsto x+a$ is an isomorphism of the Cayley
graph ${\rm Cay}(G,S)$ for every $a\in G$. Thus, in this section, we can fix
$a$ as an arbitrary nonzero element in $G$. Thus, in the sequel, we prescribe
$a$ as $(1,0,\cdots,0)\in\mathbb{F}_{2}^{n}$. Meanwhile, we can assume that
$a$ is contained in the connection set. In this section, we show that many
cubelike graphs admit FR. Our first result in this section is as follows.

######  Theorem 4.1.

Let $\Gamma={\rm Cay}(\mathbb{F}_{2^{n}},S)$ be a cubelike graph. Suppose that

$S=(0,S_{0})\cup(1,S_{1})\cup\\{(1,\mathbf{0})\\},$

where $S_{0},S_{1}\subseteq\mathbb{F}_{2}^{n-1}$, $\mathbf{0}\not\in S_{1}$.
Denote $|S_{0}|=d_{0}$ and $|S_{1}|=d_{1}$. If
$\min(v_{2}(d_{0}+d_{1}),v_{2}(d_{0}-d_{1}))\geq 3$ and $M\geq 8$, then
$\Gamma$ exhibits FR at some time $t$, where $M$ is defined by (2.16).

###### Proof.

The additive group of $\mathbb{F}_{2^{n}}$ is an elementary $2$-group
isomorphic to $\mathbb{F}_{2}^{n}$, the additive group of the $n$-dimensional
vector space over $\mathbb{F}_{2}$. Take $\mathbf{a}=(1,\mathbf{0}_{n-1})$,
where $\mathbf{0}_{n-1}\in\mathbb{F}_{2}^{n-1}$. We can assume that
$\mathbf{a}\in S$, for otherwise, we use an isomorphism of $\Gamma$ to replace
an element in $S$ with $\mathbf{a}$. Then

$G_{0}=\\{(0,\mathbf{x}):\mathbf{x}\in\mathbb{F}_{2}^{n-1}\\},\mbox{ and
}G_{1}=\\{(1,\mathbf{x}):\mathbf{x}\in\mathbb{F}_{2}^{n-1}\\},$

where $G_{0}$ and $G_{1}$ are defined by (2.10). Write
$S=(0,S_{0})\cup(1,S_{1})\cup\\{\mathbf{a}\\}$ ($0\not\in S_{1}$) and denote
$|S_{i}|=d_{i},i=0,1$. By the assumption that $|S|\equiv 1\pmod{8}$, we can
assume that $d_{0}+d_{1}+1=1+2^{\nu}d^{\prime}$, where $\nu\geq 3$. For every
$(0,\mathbf{x})\in G_{0}$, the corresponding eigenvalue is

$\lambda_{(0,\mathbf{x})}=\sum_{\mathbf{s}_{0}\in
S_{0}}(-1)^{\mathbf{s}_{0}\cdot\mathbf{x}}+\sum_{\mathbf{s}_{1}\in
S_{1}}(-1)^{\mathbf{s}_{1}\cdot\mathbf{x}}+1.$

Particularly, $\lambda_{0,\mathbf{0}}=d_{0}+d_{1}+1$. Meanwhile, for every
$(1,\mathbf{x})\in G_{1}$, the corresponding eigenvalue is

$\lambda_{(1,\mathbf{x})}=\sum_{\mathbf{s}_{0}\in
S_{0}}(-1)^{\mathbf{s}_{0}\cdot\mathbf{x}}-\sum_{\mathbf{s}_{1}\in
S_{1}}(-1)^{\mathbf{s}_{1}\cdot\mathbf{x}}+(-1).$

Particularly, $\lambda_{1,\mathbf{0}}=d_{0}-d_{1}-1$.

Since $M(\geq 8)$ is a divisor of $|\mathbb{F}_{2^{n}}|=2^{n}$ by Lemma 2.3,
we can write $M=2^{r}$, $r\geq 3$.

For a different $\mathbf{x^{\prime}}\in\mathbb{F}_{2}^{n-1}$, by the
definition of the integer $M$, it is obvious that

$\lambda_{(0,\mathbf{x})}-\lambda_{(0,\mathbf{x^{\prime}})}\equiv
0\pmod{2^{r}},\lambda_{(1,\mathbf{x})}-\lambda_{(1,\mathbf{x^{\prime}})}\equiv
0\pmod{2^{r}}.$

Write $\min(r,v_{2}(d_{0}+d_{1}),v_{2}(d_{0}-d_{1}))=\kappa$. Then $\kappa\geq
3$. Take $t=\frac{2\pi}{2^{\kappa}}$. Then

$e^{\imath
t\lambda_{x,\mathbf{y}}}=\left\\{\begin{array}[]{ll}e^{\imath\frac{2\pi}{2^{\kappa}}(d_{0}+d_{1}+1)}=e^{\imath\frac{2\pi}{2^{\kappa}}},&\mbox{
if $(x,\mathbf{y})\in G_{0}$},\\\
e^{\imath\frac{2\pi}{2^{\kappa}}(d_{0}-d_{1}-1)}=e^{-\imath\frac{2\pi}{2^{\kappa}}},&\mbox{
if $(x,\mathbf{y})\in G_{1}$}.\end{array}\right.$

Then we obtain the desired result by Theorem 2.2. ∎

Below we present a simple example as a replenishment to Theorem 4.1.

######  Example 4.2.

Let $G=\mathbb{F}_{2}^{5}$ and let the connection set $S$ be defined by

$S=\\{(10000)\\}\cup\\{(0,s):s\in S_{0}\\}\cup\\{(1,s):s\in S_{0}\\}$

where

$S_{0}=\\{(1100),(0011),(1011),(0111),(1101),(1110)\\}.$

Then $\Gamma={\rm Cay}(G,S)$ has FR at time $t=\pi/4$.

###### Proof.

Take $\mathbf{a}=(10000)$. Then

$G_{0}=\\{(0vxyz):v,x,y,z\in\\{0,1\\}\\},G_{1}=\\{(1vxyz):v,x,y,z\in\\{0,1\\}\\}.$

The eigenvalues of $\Gamma$ are

$\displaystyle\lambda_{(uxyz)}$ $\displaystyle=$
$\displaystyle(-1)^{u}+(-1)^{v+x}+(-1)^{y+z}+(-1)^{v+y+z}+(-1)^{x+y+z}+(-1)^{v+x+z}$
$\displaystyle+(-1)^{v+x+y}+(-1)^{u+v+x}+(-1)^{u+y+z}+(-1)^{u+v+y+z}$
$\displaystyle+(-1)^{u+x+y+z}+(-1)^{u+v+x+z}+(-1)^{u+v+x+y}.$

Thus

$\lambda_{(00000)}=13,\lambda_{(0vxyz)}\in\\{5,-3\\},\lambda_{(1vxyz)}=-1,v,x,y,z\in\\{0,1\\}.$

It is obvious that

$\lambda_{\mathbf{x}}\equiv\left\\{\begin{array}[]{ll}5\pmod{8},&\mbox{ if
$\mathbf{x}\in G_{0}$},\\\ -1\pmod{8},&\mbox{ if $\mathbf{x}\in
G_{1}$}.\end{array}\right.,$

Take $t=\pi/4$. Then

$e^{\imath
t\lambda_{\mathbf{x}}}=\left\\{\begin{array}[]{ll}e^{\imath\frac{\pi}{4}},&\mbox{
if $\mathbf{x}\in G_{0}$},\\\ e^{-\imath\frac{\pi}{4}},&\mbox{ if
$\mathbf{x}\in G_{1}$}.\end{array}\right.$

This completes the proof. ∎

Our next result is based on bent functions and plateaued functions over
$\mathbb{F}_{2}^{n}$. Before stating our constructions, we recall some basic
properties about such kind of functions.

Let $f:\mathbb{F}_{2}^{n}\rightarrow\mathbb{F}_{2}$ be a nonzero Boolean
function. Since the additive group $(\mathbb{F}_{2}^{n},+)$ is isomorphic to
the additive group of the finite field $\mathbb{F}_{2^{n}}$. Then $f$ viewed
as a function on $\mathbb{F}_{2^{n}}$ has a (unique) trace expansion of the
form:

$f(x)=\sum_{j\in\Gamma_{n}}{\rm
Tr}^{o(j)}_{1}(a_{j}x^{j})+\epsilon(1+x^{2^{n}-1}),\forall
x\in\mathbb{F}_{2^{n}},$

where $\Gamma_{n}$ is the set of integers obtained by choosing one element in
each cyclotomic coset of 2 modulo $2^{n}-1$, $o(j)$ is the size of the
cyclotomic coset of 2 modulo $2^{n}-1$ containing $j$,
$a_{j}\in\mathbb{F}_{2^{o(j)}}$ and $\epsilon=wt(f)$ modulo 2 where $wt(f)$ is
the Hamming weight of the image vector of $f$, that is, the number of $x$ such
that $f(x)=1$. Denote ${\rm supp}(f)=\\{x\in\mathbb{F}_{2^{n}}:f(x)=1\\}$. It
is obvious that the map $f\mapsto{\rm supp}(f)$ gives a one-to-one mapping
from the set of Boolean functions to the power set of $\mathbb{F}_{2^{n}}$.

The Walsh-Hadamard transform of $f$ is defined by

$\widehat{f}_{w}(a)=\sum_{x\in\mathbb{F}_{2^{n}}}(-1)^{f(x)+{\rm
Tr}^{n}_{1}(ax)},\forall a\in\mathbb{F}_{2^{n}}.$

######  Definition 4.3.

Let $n=2k$ with $k$ being a positive integer. A Boolean function $f$ is called
bent if $\widehat{f}_{w}(a)\in\\{\pm 2^{k}\\}$ for all
$a\in\mathbb{F}_{2^{n}}$.

It is well-known that bent function exists on $\mathbb{F}_{2^{2k}}$ for every
$k$. Moreover, for every positive integer $k$, there exist many families of
such functions. Semi-bent functions also exist in $\mathbb{F}_{2^{n}}$ for all
integer $m\geq 1$, e.g., in [32].

Let $f:\mathbb{F}_{2}^{n}\rightarrow\mathbb{F}_{2}$ be a Boolean function.
$S={\rm supp}(f)\subseteq\mathbb{F}_{2^{n}}$. For the cubelike graph
$\Gamma={\rm Cay}(\mathbb{F}_{2^{n}},S)$, its eigenvalues are
$\lambda_{0}=|S|$ and

$\lambda_{x}=\sum_{z\in
S}\chi_{x}(z)=\sum_{y\in\mathbb{F}_{2^{m}}}\frac{1-(-1)^{f(y)}}{2}\chi_{x}(y)=-\frac{1}{2}\widehat{f}_{w}(x),0\neq
x\in\mathbb{F}_{2^{n}}.$ (4.1)

For bent and semi-bent functions, the following results are known in the
literature, e.g., on pages 72 and 422 in [32].

######  Lemma 4.4.

Let $n=2k$ be a positive even integer. Let $f$ be a Boolean function on
$\mathbb{F}_{2^{n}}$ and let $S={\rm supp}(f)$.

(1) If $f$ be a bent function, then $|S|=2^{n-1}\pm 2^{k-1}$. Moreover, define
a function $\widetilde{f}$, called the dual of $f$, by

$\widehat{f}_{w}(x)=2^{k}(-1)^{\widetilde{f}(x)}.$

Then $\widetilde{f}$ is also a bent function. As a consequence, the numbers of
occurrences of $\widehat{f}_{w}(x)$ taking the values $\pm 2^{k}$ are
$2^{n-1}\pm 2^{k-1}$ or $2^{n-1}\mp 2^{k-1}$.

(2) If $f$ is a semi-bent function, then $|S|\in\\{2^{n-1},2^{n-1}\pm
2^{k-1}\\}$. Moreover, we have the following table.

Table 1. Walsh spectrum of semi-bent functions $f$ with $f(0)=0$ Value of
$\widehat{f}(x),x\in\mathbb{F}_{2^{m}}$ Frequency $0$ $2^{n-1}+2^{n-2}$
$2^{k+1}$ $2^{n-3}+2^{k-2}$ $-2^{k+1}$ $2^{n-3}-2^{k-2}$

It is known also that if $f$ is a bent function, then so is its complementary
function, i.e, $g=1+f$.

The following two constructions are actually corollaries of Theorem 3.7.

######  Corollary 4.5.

Let $n=2k\geq 4$ for a positive integer $k$ and $f$ be a bent (resp. semi-
bent) function on $\mathbb{F}_{2}^{n}$. Denote $S_{0}=supp(f)$ and define

$S=\\{(1\mathbf{0}_{n})\\}\cup\\{(0,s):\mathbf{s}\in
S_{0}\\}\cup\\{(1,s):\mathbf{s}\in S_{0}\\}\subset\mathbb{F}_{2}^{n+1}.$

Then the cubelike graph $\Gamma={\rm Cay}(\mathbb{F}_{2}^{n+1},S)$ has FR at
the time $t=\frac{\pi}{2^{k}}$ (resp. $t=\frac{\pi}{2^{k+1}}$ ).

###### Proof.

We just give the proof for the case that $f$ is a bent function. The case for
the semi-bent functions can be proved similarly. Obviously,

$G_{0}=\\{(0\mathbf{y}):\mathbf{y}\in\mathbb{F}_{2}^{n}\\},G_{1}=\\{(1\mathbf{y}):\mathbf{y}\in\mathbb{F}_{2}^{n}\\}.$

It is easily seen that the eigenvalues of $\Gamma$ are

$\lambda_{(x\mathbf{y})}=(-1)^{x}+((-1)^{x}+1)\sum_{\mathbf{s}\in
S_{0}}(-1)^{\mathbf{s}\cdot\mathbf{y}},x\in\mathbb{F}_{2},\mathbf{y}\in\mathbb{F}_{2}^{n}.$

Thus, for every $(1\mathbf{y})\in G_{1}$, $\lambda_{1\mathbf{y}}=-1$.
Moreover, for $\mathbf{y}\neq\mathbf{0}$, $(0\mathbf{y})\in G_{0}$, the
corresponding eigenvalue is

$\lambda_{(0\mathbf{y})}=1-\hat{f}_{w}(\mathbf{y})=1\pm 2^{k}.$

When $\mathbf{y}=\mathbf{0}_{n}$, the corresponding trivial eigenvalue is
$\lambda_{(0\mathbf{0}_{n})}=1+(2^{n}\pm 2^{k})$. One can check that
$\lambda_{(0\mathbf{y})}\equiv 1+2^{k}\pmod{2^{k+1}}$. Take
$t=\frac{\pi}{2^{k}}$. We have

$e^{\imath
t\lambda_{\mathbf{x}}}=\left\\{\begin{array}[]{cc}e^{\imath\frac{\pi}{2^{k}}},&\mbox{
if $\mathbf{x}\in G_{0}$},\\\ e^{-\imath\frac{\pi}{2^{k}}},&\mbox{ if
$\mathbf{x}\in G_{1}$}.\end{array}\right.$

This completes the proof. ∎

Note that when $k=2$ in the above Corollary 4.5, we can choose a bent function
$f(x_{1}x_{2}x_{3}x_{4})=x_{1}x_{2}+x_{3}x_{4}$, then we have
$supp(f)=\\{(1100),(0011),(1011),(0111),(1101),(1110)\\}$. In this case, we
get the graph in Example 4.2.

We can also use plateaued functions to construct cubelike graphs that admit
FR. The main idea is that presented in Theorem 3.7. The detail of which is
omitted.

##  5\. Concluding remarks

Using the spectral decomposition of adjacency matrices of abelian Cayley
graphs, we characterized abelian Cayley graphs having FR. This
characterization indicates potential abelian Cayley graphs admitting FR. We
established a general construction of integral graphs ${\rm
Cay}(\mathbb{Z}_{2^{s}}\times H,S)$ having FR. Meanwhile, we constructed
several new families of abelian Cayley graphs having FR. The notation of
plateaued functions over finite fields was generalized to finite abelian
groups. Based on plateaued functions, a new family of abelian Cayley graphs
having FR was built. This paper and reference [10] show that bent, semi-bent
and plateaued functions are powerful ingredients in the constructions of
abelian Cayley graphs having PST or FR. It is important to study properties of
plateaued functions over finite abelian groups. Another particularly
interesting direction is to construct Cayley graphs over non-abelian groups
having FR.

## Acknowledgement

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