# Topological phase diagram of the Haldane model
on a Bishamon-kikko–honeycomb lattice

Sogen Ikegami<EMAIL_ADDRESS>Department of Applied
Physics, The University of Tokyo, Tokyo 113-8656, Japan  Kiyu Fukui
Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan
Shun Okumura  Department of Applied Physics, The University of Tokyo, Tokyo
113-8656, Japan  Yasuyuki Kato  Department of Applied Physics, University of
Fukui, Fukui 910-8507, Japan  Yukitoshi Motome<EMAIL_ADDRESS>Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan

###### Abstract

Topological flat bands have gained extensive interest as a platform for
exploring the interplay between nontrivial band topology and correlation
effects. In recent studies, strongly correlated phenomena originating from a
topological flat band were discussed on a periodically 1/6-depleted honeycomb
lattice, but the fundamental topological nature associated with this lattice
structure remains unexplored. Here we study the band structure and topological
phase diagram for the Haldane model on this lattice, which we call the
Bishamon-kikko lattice. We also extend our study to the model connecting the
Bishamon-kikko and honeycomb lattices. We show that these models exhibit
richer topological characteristics compared to the original Haldane model on
the honeycomb lattice, such as topological insulating states with higher Chern
numbers, metallic states with nontrivial band topology even at commensurate
electron fillings, and metal-insulator transitions between them. Our findings
offer a playground of correlated topological phenomena and stimulate their
realization in a variety of two-dimensional systems, such as van der Waals
materials, graphene nanostructures, and photonic crystals.

††preprint: APS/123-QED

##  I Introduction

Topological phase of matter is one of the most exciting discoveries in modern
condensed matter physics. This concept was originally brought by the quantum
Hall effect [1, 2, 3], and later has opened a variety of exotic states, such
as topological insulators [4, 5, 6] and topological superconductors [7, 8, 9].
In these systems, the topological invariant called the Chern number $C$ [3,
10], which is given by summation of the Berry curvature [11] across the
Brillouin zone for each band, plays a pivotal role in the characterization of
the topological nature. For instance, the Haldane model on a honeycomb lattice
exhibits topological phases with $C=\pm 1$, leading to a spontaneous quantized
anomalous Hall effect [12]. Besides, flat bands, which are dispersionless
bands with a constant energy in momentum space, have also attracted much
interests. A typical example can be found in a tight-binding model on the so-
called Lieb lattice, a square lattice with $1/4$ defects, where electrons are
localized due to quantum interferences around the defects [13]. In such flat
band systems, effects of electron correlations are extremely enhanced and lead
to instabilities toward various strongly correlated phenomena, such as
ferromagnetism [14, 15, 16, 17, 18] and high-temperature superconductivity
[19, 20]. Moreover, topological flat bands, i.e., nearly flat bands with
nonzero Chern numbers, have gained significant attention since they provide a
platform for realizing the fractional quantum Hall states, where non-Abelian
fractional excitations can be utilized for topological quantum computing [21,
22, 23, 24].

Recently, a topological flat band was discussed for a honeycomb version of the
Lieb lattice obtained by periodically depleting 1/6 sites from the honeycomb
lattice [25, 26]. This lattice structure belongs to a class of superhoneycomb
systems [27], and was recently found in quenched van der Waals ferromagnet
Fe5-δGeTe2 [28, 29]. The previous studies showed that electron correlations on
this lattice can bring about orbital-selective Mott transitions and a
topological Kondo semimetal [26, 25]. Thus, this lattice has potential for
exotic phenomena associated with topological flat bands, but the fundamental
topological nature has not yet been fully explored.

In this paper, we investigate the band structure and topological nature of the
prototypical Haldane model on the $1/6$-depleted honeycomb lattice. While this
lattice structure was called a clover lattice [26], we here call it the
Bishamon-kikko (BK) lattice after Japanese traditional patterns, since the
clover lattice often refers a group of structures with $C_{4}$ or $C_{2}$
rotational symmetry; the BK lattice has $C_{3}$ rotational symmetry as shown
below. Calculating the Berry curvature for the occupied bands, we
systematically map out the topological phase diagrams while changing the model
parameters and electron filling. In addition, we consider a connection between
the BK lattice and the original honeycomb lattice by continuously changing the
transfer integrals around the defect sites, which we call the Bishamon-
kikko–honeycomb (BKH) lattice. We find that these models exhibit richer
topological properties not seen in the original Haldane model on the honeycomb
lattice, such as topological insulating states with higher Chern numbers,
metallic states with nontrivial band topology even at commensurate filling,
and metal-insulator transitions between them.

The organization of this paper is as follows. In Sec. II, we introduce our
models on the BK lattice and the BKH lattice, and the method to clarify the
topological nature of the systems. In Sec. III, we present the results for
band structures and topological phase diagrams for both models. Finally, Sec.
IV is devoted to a summary.

##  II Model and method

Figure 1:  Schematic pictures of (a) the honeycomb and (b) BK lattices
considered in the present study. The blue and yellow circles denote two
sublattices, A and B, respectively. In (b), the green arrows indicate the
primitive translation vectors, and the green rhombus shows the unit cell.
Schematics of the limits of (c) $M\rightarrow\infty$ and (d)
$M\rightarrow-\infty$ of the Haldane model on the BK lattice [Eq. (1)]. The
dashed lines represent the second-neighbor hopping surviving in each limit,
and the gray lines are guides for the eye. The limit in (c) results in a
honeycomb lattice of the second-neighbor hopping, while that in (d) is
equivalent to a kagome lattice.

###  II.1 Bishamon-kikko lattice

First, we introduce the Haldane model [12] on the BK lattice obtained by
periodically depleting $1/6$ lattice sites of the honeycomb lattice, as shown
in Figs. 1(a) and 1(b). While there are several types of the BK lattice
depending on the way of site depletion as represented by different Japanese
traditional pattens, we here focus on the type shown in Fig. 1(b). Note that
the BK lattice is similar but different from a decorated honeycomb lattice,
which is reported to have a flat band and higher-order topological insulator
phase [30, 31, 32]. We consider the Haldane model on this BK lattice, whose
Hamiltonian is given by

$\hat{H}=\sum_{i}M_{i}\hat{c}_{i}^{\dagger}\hat{c}_{i}+\left[\sum_{\langle
i,j\rangle}t_{1}\hat{c}^{\dagger}_{i}\hat{c}_{j}+\sum_{\langle\langle
i,j\rangle\rangle}t_{2}e^{i\phi_{ij}}\hat{c}^{\dagger}_{i}\hat{c}_{j}+{\rm
h.c.}\right],$ (1)

where $\hat{c}_{i}$ and $\hat{c}_{i}^{\dagger}$ are the annihilation and
creation operators of spinless fermions at site, respectively. The summations
$\langle i,j\rangle$ and $\langle\langle i,j\rangle\rangle$ are taken for all
nearest-neighbor pairs and second-neighbor pairs connected by two nearest-
neighbor bonds, respectively. Here, $M_{i}$ denotes a staggered site
potential, taking $-M$ and $+M$ on the A and B sublattices respectively
represented by the blue and yellow circles in Fig. 1(b); $t_{1}$ and $t_{2}$
represent the transfer integrals for the nearest-neighbor and second-neighbor
hopping, and $\phi_{ij}$ denotes the phase for the latter, which takes $+\phi$
and $-\phi$ for the hopping in the clockwise and counterclockwise directions
with respect to the center of each plaquette, respectively. We note that, in
the limit of $M\rightarrow\infty\ (-\infty)$, the yellow (blue) sites are
projected out, and the model ends up with the honeycomb (kagome) lattice model
composed of the second-neighbor hopping in the original model when the
electron filling is less than 2/5 (3/5), as illustrated in Fig. 1(c) [1(d)].

###  II.2 Bishamon-kikko–honeycomb lattice

Next, we introduce the Haldane model on a lattice connecting the BK and
honeycomb lattices. For this purpose, we introduce a parameter $\alpha$ that
controls the transfer integrals including defects sites on the BK lattice, and
define the Hamiltonian on the honeycomb lattice as

$\begin{split}\hat{H}=&\sum_{i}M_{i}\hat{c}_{i}^{\dagger}\hat{c}_{i}\\\
&+\left[\sum_{\langle
i,j\rangle}\alpha_{ij}t_{1}\hat{c}^{\dagger}_{i}\hat{c}_{j}+\sum_{\langle\langle
i,j\rangle\rangle}\alpha_{ij}t_{2}e^{i\phi_{ij}}\hat{c}^{\dagger}_{i}\hat{c}_{j}+{\rm
h.c.}\right],\end{split}$ (2)

where $\alpha_{ij}=\alpha$ ($0\leq\alpha\leq 1$) when the hopping process
includes a defect site on the BK lattice (for the nearest-neighbor case, when
$i$ or $j$ belongs to a defect site; for the second-neighbor case, when two
neighboring bonds consisting the second-neighbor hopping includes a defect
site), and otherwise $\alpha_{ij}=1$; the other parameters are the same as in
Eq. (1). In Eq. (2), the summations are taken on the honeycomb lattice.

The system is equivalent to the Haldane model on the honeycomb lattice when
$\alpha=1$. On the contrary, $\alpha=0$ dispatches the defect sites,
indicating that the model is reduced to Eq. (1) on the BK lattice with
isolated defects. Thus, Eq. (2) smoothly connects the two lattice systems, and
we call the lattice structure the “Bishamon-kikko–honeycomb (BKH) lattice” in
the following.

###  II.3 Method

We investigate the band structures and the topological nature of the Haldane
models on the BK lattice and the BKH lattice. To characterize the topological
nature, we calculate the summation of the Berry curvature for the occupied
states as

$C_{\rm occ}=\frac{1}{2\pi}\sum_{n,\bf{k}}^{{\rm
occ}}\left[\nabla_{\bm{k}}\times\bm{A}_{n\bm{k}}\right]_{k_{z}},$ (3)

where $\bm{A}_{n\bm{k}}=-i\expectationvalue{\nabla_{\bm{k}}}{u_{n\bm{k}}}$ is
the Berry connection; $|u_{n\bm{k}}\rangle$ denotes the eigenstate at band $n$
with momentum $\bm{k}$, and the subscript $k_{z}$ represents the $z$ component
of $\nabla_{\bm{k}}\times\bm{A}_{n\bm{k}}$. The summation in Eq. (3) is taken
for all the occupied states below the Fermi level. Thus, $C_{\rm occ}$ becomes
an integer called the Chern number $C$ when the system is insulating; a
nonzero quantized value of $C_{\rm occ}$ indicates that the system is a
topologically nontrivial insulator.

In the actual calculations of $C_{\rm occ}$, we adopt the Fukui-Hatsugai-
Suzuki method that facilitates precise numerical estimates of the Berry
curvature [33]. Dividing the hexagonal Brillouin zone into $7500$ rhombi, we
calculate the Berry curvature for each plaquette and add them up to calculate
$C_{\rm occ}$. Hereafter, we set $t_{1}=1$ as the energy scale.

##  III Results

Figure 2:  Band dispersions of the Haldane model on the BK lattice in Eq. (1)
at (a) $M=1$ and $t_{2}=0$ and (b) $M=0.3$, $t_{2}=0.3$, and $\phi=0.3\pi$.
The results are plotted along the high symmetric lines in the first Brillouin
zone shown in the inset of (a).

###  III.1 Bishamon-kikko lattice

We first examine the band dispersions and the topological phase diagrams for
the Haldane model on the BK lattice in Eq. (1). Figure 2 shows the band
dispersions along the high symmetric lines in the first Brillouin zone at (a)
$M=1$ and $t_{2}=0$ and (b) $M=0.3$, $t_{2}=0.3$, and $\phi=0.3\pi$.

When the second-neighbor hopping is absent ($t_{2}=0$), a flat band appears at
energy $E=M=1$, between the two sets of dispersive bands [Fig. 2(a)]. This is
associated with localized states at the yellow sites in Fig. 1(b), originating
from quantum interference, as found in the Lieb lattice [13]. For this
parameter set, one of the upper dispersive bands has quadratic band touching
with this flat band at the $\Gamma$ point. Each set of the dispersive bands
shows linear band crossing at the $K$ point, forming the doubly-degenerate
Dirac nodes like in the honeycomb lattice model. Meanwhile, when the second-
neighbor hopping is introduced, the flat band becomes dispersive, and the
Dirac nodes at the $K$ point are gapped out, as shown in the result for
$t_{2}=0.3$ and $\phi=0.3\pi$ in Fig. 2(b).

Figure 3:  Topological phase diagrams of the Haldane model on the BK lattice
in Eq. (1) while changing $\phi$ and $M/t_{2}$. The color contour denotes
$C_{\rm occ}$ in Eq. (3), the summation of the Berry curvature for the
occupied states. (a), (b), (c), and (d) correspond to 1/5, 2/5, 3/5, and 4/5
filling, respectively. The black lines represent the topological phase
boundaries between the Chern insulators with different integers $C_{\rm occ}$,
while the gray lines represent metal-insulator transitions where $C_{\rm occ}$
deviates from the integer values.

Figure 3 represents the topological phase diagrams at each commensurate
filling for the Haldane model on the BK lattice. The color denotes $C_{\rm
occ}$ in Eq. (3). The insulating regions with quantized integer values of
$C_{\rm occ}$ are distinguished from the metallic regions with noninteger
$C_{\rm occ}$ by the gray lines. The black lines represent the phase
boundaries between the insulators with different integers of $C_{\rm occ}$.
There are several characteristic aspects in the phase diagrams, which are not
seen in the original Haldane model on the honeycomb lattice at half filling
[12]. First of all, the phase diagrams are not symmetric with respect to the
sign of $M/t_{2}$. This is because the BK lattice has different number of
sublattice sites, unlike the honeycomb case [see Figs. 1(a) and 1(b)]. Second,
nevertheless, the phase diagrams are symmetric between $m$/5 and $1-m/5$
fillings ($m=1,2,3,4$) by changing the parameters as $\phi\rightarrow\phi+\pi$
and $M/t_{2}\rightarrow-M/t_{2}$. This is due to the particle-hole symmetry of
the model with interchanging blue/yellow sublattices in Fig. 1(b). Indeed, the
band structures are reversed in energy by the parameter changes (not shown).
Third, most notably, in addition to the wide regions of topologically
nontrivial insulators with $C_{\rm occ}=\pm 1$, which are also found in the
honeycomb case, there appear another topologically nontrivial insulating
phases with higher Chern number $C_{\rm occ}=\pm 2$. Moreover, there exist
metallic regions with metal-insulator transitions even at these commensurate
fillings. These phases with high Chern numbers as well as the metal-insulator
transitions are not found in the Haldane model on the honeycomb lattice,
indicating that the Haldane model on the BK lattice has much richer
topological phase diagrams compared with that on the honeycomb lattice.

We note that at $2/5$ ($3/5$) filling the system becomes a trivial insulator
with $C_{\rm occ}=0$ for large (small) $M/t_{2}$, as shown in Fig. 3(b)
[3(c)]. This is related to the fact that the model reduces to the honeycomb
(kagome) lattice model in the limit of $M\rightarrow\infty\ (-\infty)$, as
shown in Fig. 1(c) [1(d)]; the $2/5$ ($3/5$) filling for $M\rightarrow\infty$
corresponds to full filling of the honeycomb (kagome) model in Fig. 1(c)
[1(d)], resulting in the trivial insulating states.

###  III.2 Bishamon-kikko–honeycomb lattice

Figure 4:  Topological phase diagrams of the Haldane model on the BKH lattice
in Eq.(2) while changing $\phi$ and $\alpha$ at $M/t_{2}=0$. $\alpha$ on the
vertical axis denotes the parameter connecting the BK lattice with isolated
defects ($\alpha=0$) and the honeycomb lattice ($\alpha=1$). (a), (b), (c),
(d), and (e) correspond to 1/6, 2/6, 3/6, 4/6, 5/6 filling, respectively. The
notations are common to those in Fig. 3.  Figure 5:  Topological phase
diagrams of the Haldane model on the BKH lattice in Eq.(2) while changing
$\phi$ and $\alpha$ at $M/t_{2}=-5$. (a), (b), (c), (d), and (e) correspond to
1/6, 2/6, 3/6, 4/6, 5/6 filling, respectively. The notations are common to
those in Fig. 4.

Next we discuss the topological phase diagrams for the Haldane model on the
BKH lattice in Eq. (2). Among the model parameters $M$, $t_{2}$, $\phi$, and
$\alpha$, we display the phase diagrams at each commensurate filling at
particular values of $M/t_{2}$ with $t_{2}=0.3$ while varying $\phi$ and
$\alpha$. The results for $M/t_{2}=0$ and $M/t_{2}=-5$ are shown in Figs. 4
and 5, respectively. The notations are common to those in Fig. 3. The upper
end at $\alpha=1$ in each panel corresponds to the honeycomb lattice, while
the lower end at $\alpha=0$ corresponds to the BK lattice with isolated
defects. Thus, the results for $\alpha=1$ at half filling ($3/6$ filling)
shown in Figs. 4(c) and 5(c) reproduce those for the Haldane model on the
honeycomb lattice at half filling with corresponding values of $M/t_{2}$ [12].

Let us first focus on the results for $M/t_{2}=0$ shown in Fig. 4. As
mentioned above, the lower end at $\alpha=0$ of each panel corresponds to the
BK lattice with isolated defects; the values of $C_{\rm occ}$ at 1/6 and 5/6
fillings in Figs. 4(a) and 4(e), respectively, are the same as those for
$M/t_{2}=0$ at 1/5 and 4/5 fillings in Figs. 3(a) and 3(d). In contrast,
$C_{\rm occ}$ for $\alpha=0$ at 2/6, 3/6, and 4/6 fillings in Figs. 4(b),
4(c), and 4(d), respectively, do not match with those for $M/t_{2}=0$ in Fig.
3. This is because the flat band at $E=-M$, originating from isolated sites on
the BKH lattice, which locates at zero energy in this case, intervenes the
dispersive bands. We note that a topologically trivial insulator with $C_{\rm
occ}=0$ appears in a narrow region sandwiched by the $C_{\rm occ}=\pm 1$
phases at $2/6$ filling in Fig. 4(b). While changing $\alpha$ from $0$ to $1$,
we find that the Chern insulating phases with $C_{\rm occ}=\pm 1$ are shrunk
gradually and taken over by metallic phases through metal-insulator
transitions, except for 3/6 filling in Fig. 4(c). In this case, the insulating
phases with higher integers $|C_{\rm occ}|$ do not appear, while high values
are found in some metallic regions. We note that the model on the BKH lattice
defined by Eq. (2) has a particle-hole symmetry between $m/6$ and $1-m/6$
fillings by changing $M\rightarrow-M$ and $\phi\rightarrow\phi+\pi$, similar
to the BK lattice case; this is clearly seen in the results in Fig. 4 with
$M=0$.

Let us move on to the results for $M/t_{2}=-5$ shown in Fig. 5. As in the case
with $M/t_{2}=0$ discussed above, the values of $C_{\rm occ}$ for $\alpha=0$
at 1/6 and 5/6 fillings in Figs. 5(a) and 5(e), respectively, match with those
for $M/t_{2}=-5$ at 1/5 and 4/5 fillings in Figs. 3(a) and 3(d), while not so
at 2/6, 3/6, and 4/6 fillings, because of the intervening flat band for the
isolated sites. Since $M\neq 0$, the particle-hole symmetry is not seen in
these phase diagrams. Except for the $3/6$ and $4/6$-filling cases in Figs.
5(c) and 5(d), respectively, the Chern insulating phases with $C_{\rm occ}=\pm
1$ are shrunk and taken over by metallic regions as in the case with
$M/t_{2}=0$, but the phases with high Chern number $C_{\rm occ}=\pm 2$ appear
in the 1/6-filling case, as shown in Fig. 5(a). We note that high Chern
numbers $|C_{\rm occ}|$ are also found in the metallic regions, in particular,
at the $1/6$ and $4/6$-filling cases.

##  IV Summary

To summarize, we have studied the topological phase diagrams in the Haldane
model on the BKH lattice. Our findings reveal several characteristics that
distinguish this model from the original Haldane model on the honeycomb
lattice: topologically nontrivial insulating phases with high Chern number
$C_{\rm occ}=\pm 2$, metallic regions including high $|C_{\rm occ}|$ even at
commensurate fillings, and metal-insulator transitions between them. Our
results provide a basis for exploring intriguing phenomena induced by electron
correlations, such as novel topological phases, magnetic ordering, and
superconductivity. These lattice structures can be realized in a variety of
two-dimensional materials, such as van der Waals materials [28, 29], graphene
nanostructures [34, 35], vacancy-engineered graphene [36, 37], and photonic
crystals [38, 39]. Among them, the van der Waals materials introduced in Sec.
I may offer the BKH lattice, when the interlayer hopping can be regarded as an
effective in-plane hopping. Moreover, quantum spin systems on these lattices
would also be intriguing because of the flat bands for quasiparticle
excitations [40].

## Acknowledgments

We thank M. G. Yamada for valuable comments. This work was supported by Japan
Society for the Promotion of Science (JSPS) KAKENHI Grants No. JP23H01119 and
No. JP24K17009.

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11institutetext: Department of Mathematics, Government College University,
Faisalabad, Pakistan
E-mail<EMAIL_ADDRESS>
∗Correspondence<EMAIL_ADDRESS>
# On Topological Properties of Planar Octahedron Networks

Haidar Ali  Farzana Kousar

###### Abstract

Topological indices are scientific details of graphs which represents its
topology and of the most part graph invariant. In QSAR/QSPR, physico-chemical
characteristics and topological indices, for example, atom bond connectivity
$(ABC)$ and geometric-arithmetic $(GA)$ indices are apply to foresee the
bioactivity of concoction mixes. Graph theory discovered a significant
practice in the region of investigation. In this paper, we are taking Planar
Octahedron networks, produced by honeycomb structure of dimension $n$ and
obtain analytical closed results of Multiplicative topological indices for
firstly and presents closed formulas of degree based indices.

Mathematics Subject Classification: 05C12, 05C90

Keywords: Topological indices, Planar Octahedron network, Triangular Prism
network, Dominating Planar Octahedron network,$R_{\alpha}$, $M_{1}$, $ABC$,
$GA$, $ABC_{4}$, $GA_{5}$.

##  1 Introduction and preliminary results

Topological indices are valuable instruments that are given by graph theory.
Particles and atomic mixes are frequently demonstrated by sub-atomic graph. In
perspective on graph hypothesis, a sub-atomic chart depicts the auxiliary
recipe of synthetic compound by changing over molecules by vertices and
concoction bonds by edges. The blend of chemical science, arithmetic and data
science is a cheminformatics. It predicts the property of chemical compound by
using Quantitative structure-activity (QSAR)and structure property (QSPR)
relationships. The topological indices are used to predict chemical compound
bioactivity.

A representation of numbers, polynomials and matrices are representations of a
graph. Graph has its own characteristics which can be determined by
topological indices and the topology of graph remains unchanged under
automorphism of graph. In the different classes of indices, degree based
topological indices are of extraordinary significance and assume an essential
job in substance chart hypothesis and especially in science. In increasingly
exact manner, a topological index $Top(H)$ of a graph, is a number with the
property that for each graph $G$ isomorphic to $H$, $Top(H)=Top(G)$. The idea
of topological index originated from Wiener [20] while he was dealing with
boiling point of paraffin, named this record as _path number_. Later on,
renamed as _Wiener index_ [5].

Drawing Algorithm for Planar Octahedron (POH) Networks
Step 1: Construct a Silicate network of dimension n.
Step 2: Fix new vertices at the centroid of each triangular face and connect
it to oxide vertices in the corresponding triangle face.
Step 3: Connect all these new centroid vertices which lie in the same silicate
cell.
Step 4: Remove all Silicon vertices. The resulting graph is called planar
Octahedron network of dimension n.
Also we can derive Triangular Prism network $TP$ and Dominating Planar
Octahedron network $DPOH$.

Figure 1: Planar Octahedron network $POH(2)$.
Figure 2: Triangular Prism network $TP(2)$.
Figure 3: Dominating Planar Octahedron network $DPOH(2)$.

In this article, $G$ is considered to be a network with vertex set $V(G)$ and
edge set $E(G)$, the $deg(u)$ is the degree of vertex $u\in V(G)$ and
$S_{u}={\sum\limits_{v\in N_{G}(u)}deg(v)}$ where $N_{G}(u)=\\{v\in V(G)\mid
uv\in E(G)\\}$. The notations used in this article are mainly taken from the
books [6, 10].

Let $G$ be a connected graph. Then the Wiener index of $G$ is defined as

$W(G)=\frac{1}{2}\sum\limits_{(u,v)}d(u,v),$ (1)

where $(u,v)$ is any ordered pair of vertices in $G$ and $d(u,v)$ is $u-v$
geodesic.

The very first and oldest degree based topological index is _Randić_ index
[15] denoted by $R_{-\frac{1}{2}}(G)$ and was introduced by Milan Randić and
is defined as

$R_{-\frac{1}{2}}(G)=\sum\limits_{uv\in E(G)}\frac{1}{\sqrt{deg(u)deg(v)}}.$
(2)

The general Randić index $R_{\alpha}(G)$ is the sum of
$(deg(u)deg(v))^{\alpha}$ over all the edges $e=uv\in E(G)$ and is defined as

$R_{\alpha}(G)=\sum\limits_{uv\in E(G)}(deg(u)deg(v))^{\alpha}\,\,\
\text{for}\,\,\ \alpha=1,\frac{1}{2},-1,-\frac{1}{2}.$ (3)

An important topological index introduced by Ivan Gutman and Trinajstić is the
_Zagreb_ index denoted by $M_{1}(G)$ and is defined as

$M_{1}(G)=\sum_{uv\in E(G)}(deg(u)+deg(v)).$ (4)

One of the well-known degree based topological index is _atom-bond
connectivity_ $(ABC)$ index introduced by Estrada _et al._ in [7] and defined
as

$ABC(G)=\sum_{uv\in E(G)}\sqrt{\frac{deg(u)+deg(v)-2}{deg(u)deg(v)}}.$ (5)

Another well-known connectivity topological descriptor is _geometric-
arithmetic_ $(GA)$ index which was introduced by Vukičević _et al._ in [18]
and defined as

$GA(G)=\sum_{uv\in E(G)}\frac{2\sqrt{deg(u)deg(v)}}{(deg(u)+deg(v))}.$ (6)

The fourth version of atom-bond connectivity index denoted by $ABC_{4}$ and
fifth version of geometric-arithmetic index denoted by $GA_{5}$ can be
computed if we are able to find the edge partition of these interconnection
networks based on sum of the degrees of end vertices of each edge in these
graphs. The fourth version of $ABC$ index is introduced by Ghorbani _et al._
[12] and defined as

$ABC_{4}(G)=\sum_{uv\in E(G)}\sqrt{\frac{S_{u}+S_{v}-2}{S_{u}S_{v}}}.$ (7)

Recently, the fifth version of $GA$ index is proposed by Graovac _et al._ [13]
and is defined as

$GA_{5}(G)=\sum_{uv\in E(G)}\frac{2\sqrt{S_{u}S_{v}}}{(S_{u}+S_{v})}.$ (8)

The general Randić index for $\alpha=1$ is the second Zagreb index for any
graph $G$.

##  2 Main results

In this article, we study the general Randić, first Zagreb, $ABC$, $GA$,
$ABC_{4}$ and $GA_{5}$ indices and give exact results of these indices for
Planar Octahedron $POH(n)$ network, Triangular Prism $TP(n)$ network and
Dominating Planar Octahedron $DPOH(n)$ network. Nowadays there is an extensive
research activity on $ABC$ and $GA$ indices and their variants, for further
study of topological indices of various graphs, [1, 2, 3, 4, 9, 14, 16, 17,
19].

###  2.1 Results for Planar Octahedron $POH(n)$ networks

In this section, we calculate certain degree based topological indices of
Planar Octahedron $POH(n)$ network. We compute general Randić $R_{\alpha}(G)$
with $\alpha=\\{1,-1,\frac{1}{2},-\frac{1}{2}\\}$, $ABC$, $GA$, $ABC_{4}$ and
$GA_{5}$ indices for $POH(n)$ network.

###### Theorem 2.1.1

Consider the Planar Octahedron network $G\cong POH(n)$ for $n\in\mathbb{N}$.
Then its general Randić index is equal to

$R_{\alpha}(POH(n))=\left\\{\begin{array}[]{ll}288n(-2+9n),&\hbox{$\alpha=1$;}\\\
24n(-2+(9+6\sqrt{2})n),&\hbox{$\alpha=\frac{1}{2}$;}\\\
\frac{9}{32}n(2+9n),&\hbox{$\alpha=-1$;}\\\
\frac{3}{4}n(2+(9+6\sqrt{2})n),&\hbox{$\alpha=-\frac{1}{2}$.}\end{array}\right.$

###### Proof

Let $G$ be the Planar Octahedron network. The Planar Octahedron network
$POH(n)$ has $27n^{2}-21n$ vertices and the edge set of $POH(n)$ is divided
into three partitions based on the degree of end vertices. Table 1 shows such
an edge partition of $POH(n)$. Thus from $(3)$ is follows that,

$R_{\alpha}(G)=\sum\limits_{uv\in E(G)}(deg(u)deg(v))^{\alpha}.$

For $\alpha=1$
Now we apply the formula,

$R_{1}(G)=\sum_{uv\in E_{j}(G)}\sum_{j=1}^{3}deg{(u)}\cdot deg{(v)}.$

By using edge partition given in Table 1, we get

$R_{1}(G)=16|E_{1}(POH(n))|+32|E_{2}(POH(n))|+64|E_{3}(POH(n))|,$
$\Longrightarrow R_{1}(G)=288n(-2+9n).$

For $\alpha=\frac{1}{2}$
We apply the formula,

$R_{\frac{1}{2}}(G)=\sum_{uv\in E_{j}(G)}\sum_{j=1}^{3}\sqrt{deg(u)\cdot
deg(v)}.$

By using edge partition given in Table 1, we get

$R_{\frac{1}{2}}(G)=4|E_{1}(POH(n))|+4\sqrt{2}|E_{2}(POH(n))|+8|E_{3}(POH(n))|,$
$\Longrightarrow R_{\frac{1}{2}}(G)=24n(-2+(9+6\sqrt{2})n).$

For $\alpha=-1$
We apply the formula,

$R_{-1}(G)=\sum_{uv\in E_{j}(G)}\sum_{j=1}^{3}\frac{1}{deg(u)\cdot deg(v)}.$

$R_{-1}(G)=\frac{1}{16}|E_{1}(POH(n))|+\frac{1}{32}|E_{2}(POH(n))|+\frac{1}{64}|E_{3}(POH(n))|,$

By using edge partition given in Table 1, we get

$\Longrightarrow R_{-1}(G)=\frac{9}{32}n(2+9n).$

For $\alpha=-\frac{1}{2}$
We apply the formula,

$R_{-\frac{1}{2}}(G)=\sum_{uv\in
E_{j}(G)}\sum_{j=1}^{3}\frac{1}{\sqrt{deg(u).\cdot deg(v)}}.$
$R_{-\frac{1}{2}}(G)=\frac{1}{4}|E_{1}(POH(n))|+\frac{1}{4}{\sqrt{2}}|E_{2}(POH(n))|+\frac{1}{8}|E_{3}(POH(n))|,$

By using edge partition given in Table 1, we get

$\Longrightarrow R_{-\frac{1}{2}}(G)=\frac{3}{4}n(2+(9+6\sqrt{2})n).$

In the following theorem, we compute first Zagreb index of Planar Octahedron
network $POH(n)$.

###### Theorem 2.1.2

For Planar Octahedron network $G\cong POH(n)$ for $n\in\mathbb{N}$. Then its
first Zagreb index is equal to

$M_{1}(POH(n))=96n(-1+9n).$

###### Proof

Let $G$ be the Planar Octahedron network $POH(n)$. By using edge partition
from Table 1, the result follows.

$M_{1}(POH(n))=\sum_{uv\in E(G)}(deg(u)+deg(v))=\sum_{uv\in
E_{j}(G)}\sum_{j=1}^{3}(deg(u)+deg(v)).$
$M_{1}(POH(n))=8|E_{1}(POH(n))|+12|E_{2}(POH(n))|+16|E_{3}(POH(n))|,$

By doing some calculation, we get

$\Longrightarrow M_{1}(POH(n))=96n(-1+9n).$

Now, we compute $ABC$, $GA$, $ABC_{4}$ and $GA_{5}$ indices of Planar
Octahedron network $POH(n)$.

###### Theorem 2.1.3

Let $D\cong POH(n)$ be the Planar Octahedron network, then

$\bullet$ $ABC(G)$ =
$\frac{3}{4}n(4\sqrt{6}-2\sqrt{14}+3(4\sqrt{5}+2\sqrt{6}+\sqrt{14})n)$, for
$n\in\mathbb{N}$

$\bullet$ $GA(G)$ = $12(3+2\sqrt{2})n^{2}$, for $n\in\mathbb{N}$

$\bullet$ $ABC_{4}(G)$ =
$\frac{1}{440}(264\sqrt{29}+88\sqrt{930}+24\sqrt{2255}+5280(-1+n)+60\sqrt{330}(-1+n)n+24\sqrt{3410}(-1+n)+165\sqrt{94}(-1+n)^{2}+528\sqrt{35}n+132\sqrt{38}n+330\sqrt{46}(-1+n)n+60\sqrt{86}(-3+2n)+220\sqrt{35}(2-5n+3n^{2}))$,
for $n\geq 2$

$\bullet$
$GA_{5}$=$8\sqrt{2}+6\sqrt{15}+\frac{8}{7}\sqrt{110}+\frac{48}{23}\sqrt{33}(-1+n)+\frac{3}{2}\sqrt{55}(-1+n)+\frac{96}{17}\sqrt{66}(-1+n)-36n+\frac{48}{11}\sqrt{30}n+36n^{2}+8\sqrt{2}(2-5n+3n^{2})$,
for $n\geq 2$

###### Proof

By using edge partition given in Table 1, we get the result. From $(5)$ it
follows that,

$ABC(G)=\sum_{uv\in E(G)}\sqrt{\frac{deg(u)+deg(v)-2}{deg(u)\cdot
deg(v)}}=\sum_{uv\in
E_{j}(G)}\sum_{j=1}^{3}\sqrt{\frac{deg(u)+deg(v)-2}{deg(u)\cdot deg(v)}}.$
$ABC(G)=\frac{\sqrt{6}}{4}|E_{1}(POH(n))|+\dfrac{\sqrt{5}}{4}|E_{2}(POH(n))|+\dfrac{\sqrt{14}}{8}|E_{3}(POH(n))|,$

By doing some calculation, we get

$\Longrightarrow
ABC(G)=\frac{3}{4}n(4\sqrt{6}-2\sqrt{14}+3(4\sqrt{5}+2\sqrt{6}+\sqrt{14})n).$

from (6) we get

$GA(G)=\sum_{uv\in
E(G)}\frac{2\sqrt{deg(u)deg(v)}}{(deg(u)+deg(v))}=\sum_{j=1}^{3}\sum_{uv\in
E_{j}(G)}\frac{2\sqrt{deg(u)deg(v)}}{(deg(u)+deg(v))}.$

By doing some calculation, we get

$GA(G)=|E_{1}(POH(n))|+\frac{2\sqrt{2}}{3}|E_{2}(POH(n))|+|E_{3}(POH(n))|,$
$\Longrightarrow GA=\frac{4}{3}n((9+6\sqrt{2})n+2\sqrt{2}-3)$

Let us consider an edge partition based on degree sum of neighbors of end
vertices. Then the edge set $E(POH(n))$ can be divided into twelve edge
partitions $E_{j}(POH(n)),4\leq j\leq 15$. Table 2 shows such edge partitions.
From (7) we get

$ABC_{4}(G)=\sum_{uv\in
E(G)}\sqrt{\frac{S_{u}+S_{v}-2}{S_{u}S_{v}}}=\sum_{j=4}^{15}\sum_{uv\in
E_{j}(G)}\sqrt{\frac{S_{u}+S_{v}-2}{S_{u}S_{v}}}.$
$\displaystyle ABC_{4}(G)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{38}}{20}|E_{4}(POH(n))|+\frac{\sqrt{35}}{20}|E_{5}(POH(n))|+\frac{\sqrt{29}}{20}|E_{6}(POH(n))|+$
$\displaystyle 0.2654|E_{7}(POH(n))|+\frac{\sqrt{46}}{4}|E_{8}(POH(n))|+$
$\displaystyle 0.2541|E_{9}(POH(n))|+\frac{1}{4}|E_{10}(POH(n))|+$
$\displaystyle\frac{\sqrt{35}}{24}|E_{11}(POH(n))|+0.2518|E_{12}(POH(n))|+$
$\displaystyle\frac{\sqrt{86}}{44}|E_{13}(POH(n))|+\frac{\sqrt{330}}{88}|E_{14}(POH(n))|+$
$\displaystyle\frac{\sqrt{94}}{48}|E_{15}(POH(n))|$

$\Longrightarrow ABC_{4}$ =
$\frac{1}{440}(264\sqrt{29}+88\sqrt{930}+24\sqrt{2255}+5280(-1+n)+60\sqrt{330}(-1+n)n+24\sqrt{3410}(-1+n)+165\sqrt{94}(-1+n)^{2}+528\sqrt{35}n+132\sqrt{38}n+330\sqrt{46}(-1+n)n+60\sqrt{86}(-3+2n)+220\sqrt{35}()2-5n+3n^{2})$
and from (8) we get

$GA_{5}(G)=\sum_{uv\in
E(G)}\frac{2\sqrt{S_{u}S_{v}}}{(S_{u}+S_{v})}=\sum_{j=4}^{15}\sum_{uv\in
E_{j}(G)}\frac{2\sqrt{S_{u}S_{v}}}{(S_{u}+S_{v})}.$

$GA_{5}(G)=2\bigg{[}|E_{4}(POH(n))|+\frac{2\sqrt{30}}{14}|E_{5}(POH(n))|+\frac{2\sqrt{2}}{3}|E_{6}(POH(n))|+\\\
\frac{\sqrt{55}}{8}|E_{7}(POH(n))|+|E_{8}(POH(n))|+\frac{\sqrt{15}}{4}|E_{9}(POH(n))|+\frac{2\sqrt{66}}{7}|E_{10}(POH(n))|+\frac{2\sqrt{2}}{3}|E_{11}(POH(n))|+\frac{2\sqrt{110}}{21}|E_{12}(POH(n))|+|E_{13}(POH(n))|+\frac{4\sqrt{33}}{23}|E_{14}(POH(n))|+|E_{1}5(POH(n))|\bigg{]}$
$\Longrightarrow GA_{5}$ =
$8\sqrt{2}+6\sqrt{15}+\frac{8}{7}\sqrt{110}+\frac{48}{23}\sqrt{33}(-1+n)+\frac{3}{2}\sqrt{55}(-1+n)+\\\
\frac{96}{17}\sqrt{66}(-1+n)-36n+\frac{48}{11}\sqrt{30}n+36n^{2}+8\sqrt{2}(2-5n+3n^{2})$.

###  2.2 Results for Triangular Prism network $TP(n)$

In this section, we calculate certain degree based topological indices of
Triangular Prism network $TP(n)$ of dimension $n$. In the coming theorems we
compute general Randić index $R_{\alpha}(G)$ with
$\alpha=\\{1,-1,\frac{1}{2},-\frac{1}{2}\\}$, $ABC$, $GA$, $ABC_{4}$ and
$GA_{5}$ of $TP(n)$.

###### Theorem 2.2.1

Consider the Triangular Prism TP(n) network $G\cong TP(n)$ for
$n\in\mathbb{N}$. Then its general Randić index is equal to

$R_{\alpha}(TP(n))=\left\\{\begin{array}[]{ll}54n(-5+21n),&\hbox{$\alpha=1$;}\\\
18n(-3+\sqrt{2}+3(3+\sqrt{2})n),&\hbox{$\alpha=\frac{1}{2}$;}\\\
\frac{1}{6}n(4+21n),&\hbox{$\alpha=-1$;}\\\
n(\sqrt{2}+3(3+\sqrt{2})n),&\hbox{$\alpha=-\frac{1}{2}$.}\end{array}\right.$

###### Proof

Let $G$ be the Triangular Prism network. The Triangular Prism network $TP(n)$
has $27n^{2}+3n$ vertices and the set of edges $TP(n)$ divided into three
partitions based on the degree of end vertices. Table 3 shows such an edge
partition of $TP(n)$. Thus from $(3)$ is follows that

$R_{\alpha}(G)=\sum\limits_{uv\in E(G)}(deg(u)deg(v))^{\alpha}.$

Now we apply the formula,

$R_{1}(G)=\sum_{uv\in E_{j}(G)}\sum_{j=1}^{3}deg{(u)}\cdot deg{(v)}$

By using edge partition given in Table 3, we get

$R_{1}(G)=9|E_{1}(TP(n))|+18|E_{2}(TP(n))|+36|E_{3}(TP(n))|$ $\Longrightarrow
R_{1}(G)=54n(-5+21n).$

For $\alpha=\frac{1}{2}$
We apply the formula,

$R_{\frac{1}{2}}(G)=\sum_{uv\in E_{j}(G)}\sum_{j=1}^{3}\sqrt{deg(u)\cdot
deg(v)}.$

By using edge partition given in Table 3, we get

$R_{\frac{1}{2}}(G)=3|E_{1}(TP(n))|+3\sqrt{2}|E_{2}(TP(n))|+6|E_{3}(TP(n))|$
$\Longrightarrow R_{\frac{1}{2}}(G)=18n(-3+\sqrt{2}+3(3+\sqrt{2})n).$

For $\alpha=-1$
We apply the formula,

$R_{-1}(G)=\sum_{uv\in E_{j}(G)}\sum_{j=1}^{3}\frac{1}{deg(u)\cdot deg(v)}.$

$R_{-1}(G)=\frac{1}{9}|E_{1}(TP(n))|+\frac{1}{18}|E_{2}(TP(n))|+\frac{1}{36}|E_{3}(TP(n))|$
$\Longrightarrow R_{-1}(G)=\frac{1}{6}n(4+21n).$

For $\alpha=-\frac{1}{2}$
We apply the formula,

$R_{-\frac{1}{2}}(G)=\sum_{uv\in
E_{j}(G)}\sum_{j=1}^{3}\frac{1}{\sqrt{deg(u)\cdot deg(v)}}.$
$R_{-\frac{1}{2}}(G)=\frac{1}{23}|E_{1}(TP(n))|+\frac{1}{3\sqrt{2}}|E_{2}(TP(n))|+\frac{1}{6}|E_{3}(TP(n))|$
$\Longrightarrow R_{-\frac{1}{2}}(G)=n(\sqrt{2}+3(3+\sqrt{2})n).$

In the following theorem, we compute first Zagreb index of Triangular Prism
network $TP(n)$.

###### Theorem 2.2.2

For Triangular Prism network $G\cong TP(n)$ for $n\in\mathbb{N}$. Then its
first Zagreb index is equal to

$M_{1}(TP(n))=54n(-1+9n).$

###### Proof

Let $G$ be the Triangular Prism network $TP(n)$. By using edge partition from
Table 3, the result follows. From (4) we have

$M_{1}(TP(n))=\sum_{uv\in E(G)}(deg(u)+deg(v))=\sum_{uv\in
E_{j}(G)}\sum_{j=1}^{3}(deg(u)+deg(v)).$
$M_{1}(TP(n))=6|E_{1}(TP(n))|+9|E_{2}(TP(n))|+12|E_{3}(TP(n))|$

By doing some calculation, we get

$\Longrightarrow M_{1}(TP(n))=54n(-1+9n).$

Now, we compute $ABC$, $GA$, $ABC_{4}$ and $GA_{5}$ indices of Triangular
Prism network $TP(n)$.

###### Theorem 2.2.3

Let $G\cong TP(n)$ be the Triangular Prism network, then

$ABC(G)$ = $n(4-2\sqrt{10}+\sqrt{14}+3(4+\sqrt{10}+\sqrt{14})n)$, for $n\geq
1$

$GA(G)$ = $2n(-3+2\sqrt{2}+6(3+\sqrt{2})n)$, for $n\geq 1$

$ABC_{4}(G)$ =
$4-\frac{\sqrt{2}}{3}-\frac{4\sqrt{13}}{3}+\sqrt{\frac{74}{5}}+\sqrt{17}-\frac{5\sqrt{22}}{3}-\frac{4\sqrt{37}}{3}+\frac{3\sqrt{58}}{5}+\\\
\frac{2}{45}(-225+60\sqrt{2}+20\sqrt{13}+15\sqrt{22}+30\sqrt{37}+15\sqrt{57}-\\\
27\sqrt{58}+3\sqrt{330})n+[6+3\sqrt{\frac{11}{2}}+\frac{3\sqrt{58}}{5}]n^{2}$,
for $n\geq 2$

$GA_{5}(G)$ =
$-\frac{444}{13}+\frac{280\sqrt{2}}{17}-\frac{36\sqrt{5}}{7}+\frac{5760\sqrt{10}}{1729}+[-\frac{24}{13}+\frac{48\sqrt{3}}{7}+\frac{36\sqrt{5}}{7}-\\\
\frac{636\sqrt{10}}{133}+\frac{3\sqrt{15}}{2}]n+\frac{36}{7}(7+\sqrt{10})n^{2}$,
for $n\geq 2$

###### Proof

By using edge partition given in Table 3 , we get the result. From $(5)$ it
follows that

$ABC(G)=\sum_{uv\in E(G)}\sqrt{\frac{deg(u)+deg(v)-2}{deg(u)\cdot
deg(v)}}=\sum_{uv\in
E_{j}(G)}\sum_{j=1}^{3}\sqrt{\frac{deg(u)+deg(v)-2}{deg(u)\cdot deg(v)}}.$
$ABC(G)=\frac{2}{3}|E_{1}(TP(n))|+\frac{\sqrt{14}}{6}|E_{2}(TP(n))|+\frac{\sqrt{10}}{6}|E_{3}(TP(n))|$

By doing some calculation, we get

$\Longrightarrow ABC(G)=n(4-2\sqrt{10}+\sqrt{14}+3(4+\sqrt{10}+\sqrt{14})n).$

from (6) we get

$GA(G)=\sum_{uv\in
E(G)}\frac{2\sqrt{deg(u)deg(v)}}{(deg(u)+deg(v))}=\sum_{uv\in
E_{j}(G)}\sum_{j=1}^{3}\frac{2\sqrt{deg(u)deg(v)}}{(deg(u)+deg(v))}.$

By doing some calculation, we get

$GA(G)=|E_{1}(TP(n))|+\frac{2\sqrt{2}}{3}|E_{2}(TP(n))|+|E_{3}(TP(n))|$
$\Longrightarrow GA=2n(-3+2\sqrt{2}+6(3+\sqrt{2})n).$

Let us consider an edge partition based on degree sum of neighbors of end
vertices. The edge set $E(TP(n))$ divided into twelve edge partitions
$E_{j}(TP(n)),\\\ 4\leq j\leq 15$, where the edge partition $E_{4}(TP(n))$
contains $12n$ edges $uv$ with $S_{u}=9andS_{v}=12$, the edge partition
$E_{5}(TP(n))$ contains $6n$ edges $uv$ with $S_{u}=9$ and $S_{v}=15$, the
edge partition $E_{6}(TP(n))$ contains $18n^{2}-12$ edges $uv$ with
$S_{u}=S_{v}=12$, the edge partition $E_{7}(TP(n))$ contains $12$ edges $uv$
with $S_{u}=12$ and $S_{v}=24$, the edge partition $E_{8}(TP(n))$ contains
$24n-24$ edges $uv$ with $S_{u}=12$ and $S_{v}=27$, the edge partition
$E_{9}(TP(n))$ contains $18n^{2}-30n+12$ edges $uv$ with $S_{u}=12$ and
$S_{v}=30$, the edge partition $E_{10}(TP(n))$ contains $12$ edges $uv$ with
$S_{u}=15$ and $S_{v}=24$, the edge partition $E_{11}(TP(n))$ contains
$12n-12$ edges $uv$ with $S_{u}=15$ and $S_{v}=27$, the edge partition
$E_{12}(TP(n))$ contains $12$ edges $uv$ with $S_{u}=24$ and $S_{v}=27$, the
edge partition $E_{13}(TP(n))$ contains $12n-18$ edges $uv$ with
$S_{u}=S_{v}=27$, the edge partition $E_{14}(TP(n))$ contains $12n-12$ edges
$uv$ with $S_{u}=27$ and $S_{v}=30$ and the edge partition $E_{15}(TP(n))$
contains $18n^{2}-36n+18$ edges $uv$ with $S_{u}=S_{v}=30$.
From (7) we get

$ABC_{4}(G)=\sum_{uv\in
E(G)}\sqrt{\frac{S_{u}+S_{v}-2}{S_{u}S_{v}}}=\sum_{j=4}^{15}\sum_{uv\in
E_{j}(G)}\sqrt{\frac{S_{u}+S_{v}-2}{S_{u}S_{v}}}.$

$ABC_{4}(G)=\frac{\sqrt{57}}{18}|E_{4}(TP(n))|+\frac{\sqrt{330}}{45}|E_{5}(TP(n))|+\frac{\sqrt{22}}{12}|E_{6}(TP(n))|+\\\
\frac{\sqrt{17}}{12}|E_{7}(TP(n))|+\frac{\sqrt{37}}{18}|E_{8}(TP(n))|+\frac{1}{3}|E_{9}(TP(n))|+\frac{\sqrt{370}}{60}|E_{10}(TP(n))|+\frac{2\sqrt{2}}{9}|E_{11}(TP(n))|+\frac{7\sqrt{2}}{36}|E_{12}(TP(n))|+\frac{2\sqrt{13}}{27}|E_{13}(TP(n))|+\\\
\frac{\sqrt{22}}{18}|E_{14}(TP(n))|+\frac{\sqrt{58}}{30}|E_{15}(TP(n))|$
$\Longrightarrow ABC_{4}$ =
$4-\frac{\sqrt{2}}{3}-\frac{4\sqrt{13}}{3}+\sqrt{\frac{74}{5}}+\sqrt{17}-\frac{5\sqrt{22}}{3}-\frac{4\sqrt{37}}{3}+\frac{3\sqrt{58}}{5}+\\\
\frac{2}{45}(-225+60\sqrt{2}+20\sqrt{13}+15\sqrt{22}+30\sqrt{37}+15\sqrt{57}-\\\
27\sqrt{58}+3\sqrt{330})n+[6+3\sqrt{\frac{11}{2}}+\frac{3\sqrt{58}}{5}]n^{2}$
and from (8) we get

$GA_{5}(G)=\sum_{uv\in
E(G)}\frac{2\sqrt{S_{u}S_{v}}}{(S_{u}+S_{v})}=\sum_{j=4}^{15}\sum_{uv\in
E_{j}(G)}\frac{2\sqrt{S_{u}S_{v}}}{(S_{u}+S_{v})}.$

$GA_{5}(G)=\frac{4\sqrt{3}}{7}|E_{4}(TP(n))|+\frac{\sqrt{15}}{4}|E_{5}(TP(n))|+|E_{6}(TP(n))|+\\\
\frac{2\sqrt{2}}{3}|E_{7}(TP(n))|+\frac{12}{13}|E_{8}(TP(n))|+\frac{2\sqrt{10}}{7}|E_{9}(TP(n))|+\\\
\frac{4\sqrt{10}}{13}|E_{10}(TP(n))|+\frac{3\sqrt{5}}{7}|E_{11}(TP(n))|+\frac{12\sqrt{2}}{17}|E_{12}(TP(n))|+\\\
|E_{13}(TP(n))|+\frac{6\sqrt{10}}{13}|E_{19}(TP(n))|+|E_{15}(TP(n))|$
$\Longrightarrow GA_{5}$ =
$-\frac{444}{13}+\frac{280\sqrt{2}}{17}-\frac{36\sqrt{5}}{7}+\frac{5760\sqrt{10}}{1729}+[-\frac{24}{13}+\frac{48\sqrt{3}}{7}+\frac{36\sqrt{5}}{7}-\\\
\frac{636\sqrt{10}}{133}+\frac{3\sqrt{15}}{2}]n+\frac{36}{7}(7+\sqrt{10})n^{2}$.

Now, we calculate certain degree based topological indices of Dominating
Planar Octahedral network $DPOH(n)$. In the coming theorems we compute general
Randić index $R_{\alpha}(G)$ with
$\alpha=\\{1,-1,\frac{1}{2},-\frac{1}{2}\\}$, $ABC$, $GA$, $ABC_{4}$ and
$GA_{5}$ of $DPOH(n)$.

###### Theorem 2.2.4

Consider the Dominating Planar Octahedral network $G\cong DPOH(n)$ for
$n\in\mathbb{N}$. Then its general Randić index is equal to

$R_{\alpha}(RTSL(n))=\left\\{\begin{array}[]{ll}288(11-31n+27n^{2}),&\hbox{$\alpha=1$;}\\\
24(11+6\sqrt{2}-(31+18\sqrt{2})n+9(3+2\sqrt{2})n^{2}),&\hbox{$\alpha=\frac{1}{2}$;}\\\
\frac{9}{32}(7-23n+27n^{2}),&\hbox{$\alpha=-1$;}\\\
\frac{3}{4}(7+6\sqrt{2}-(23+18\sqrt{2})n+9(3+2\sqrt{2})n^{2}),&\hbox{$\alpha=-\frac{1}{2}$.}\end{array}\right.$

###### Proof

Let $G$ be the Dominating Planar Octahedral network. The Dominating Planar
Octahedral network $DPOH(n)$ has $81^{2}-75n+24$ vertices. And the edges of
$DPOH(n)$ has $216^{2}-216+72$. The edge set of $DPOH(n)$ is divided into
three partitions based on the degree of end vertices. The first edge partition
$E_{1}(DPOH(n))$ contains $54n^{2}-30n+6$ edges $uv$, where $deg(u)=deg(v)=4$.
The second edge partition $E_{2}(DPOH(n))$ contains $108^{2}-108n+36$ edges
$uv$, where $deg(u)=4$ and $deg(v)=8$. The third edge partition
$E_{3}(DPOH(n))$ contains $54n^{2}-78n+30$ edges $uv$, where
$deg(u)=deg(v)=8$. Table 5 shows such an edge partition of $DPOH(n)$. Thus
from $(3)$ is follows that

$R_{\alpha}(G)=\sum\limits_{uv\in E(G)}(deg(u)deg(v))^{\alpha}.$

Now we apply the formula,

$R_{1}(G)=\sum_{uv\in E_{j}(G)}\sum_{j=1}^{3}deg{(u)}\cdot deg{(v)}.$

By using edge partition given in Table 5, we get

$R_{1}(G)=16|E_{1}(DPOH(n))|+32|E_{2}(DPOH(n))|+64|E_{3}(DPOH(n))|$
$\Longrightarrow R_{1}(G)=288(11-31n+27n^{2}).$

For $\alpha=\frac{1}{2}$
We apply the formula,

$R_{\frac{1}{2}}(G)=\sum_{uv\in E_{j}(G)}\sum_{j=1}^{3}\sqrt{deg(u)\cdot
deg(v)}.$

By using edge partition given in Table 5, we get

$R_{\frac{1}{2}}(G)=4|E_{1}(DPOH(n))|+4\sqrt{2}|E_{2}(DPOH(n))|+8|E_{3}(DPOH(n))|$
$\Longrightarrow
R_{\frac{1}{2}}(G)=24(11+6\sqrt{2}-(31+18\sqrt{2})n+9(3+2\sqrt{2})n^{2}).$

For $\alpha=-1$
We apply the formula,

$R_{-1}(G)=\\\ sum_{uv\in E_{j}(G)}\sum_{j=1}^{3}\frac{1}{deg(u)\cdot
deg(v)}.$

$R_{-1}(G)=\frac{1}{16}|E_{1}(DPOH(n))|+\frac{1}{32}|E_{2}(DPOH(n))|+\frac{1}{64}|E_{3}(DPOH(n))|$
$\Longrightarrow R_{-1}(G)=\frac{9}{32}(7-23n+27n^{2}).$

For $\alpha=-\frac{1}{2}$
We apply the formula,

$R_{-\frac{1}{2}}(G)=\sum_{uv\in
E_{j}(G)}\sum_{j=1}^{3}\frac{1}{\sqrt{deg(u)\cdot deg(v)}}.$
$R_{-\frac{1}{2}}(G)=\frac{1}{4}|E_{1}(DPOH(n))|+\frac{1}{4\sqrt{2}}|E_{2}(DPOH(n))|+\frac{1}{8}|E_{3}(DPOH(n))|$
$\Longrightarrow
R_{-\frac{1}{2}}(G)=\frac{3}{4}(7+6\sqrt{2}-(23+18\sqrt{2})n+9(3+2\sqrt{2})n^{2}).$

In the following theorem, we compute first Zagreb index of Dominating Planar
Octahedral network $DPOH(n)$.

###### Theorem 2.2.5

For Dominating Planar Octahedral network $DPOH(n)$, the first Zagreb index is
equal to

$M_{1}(DPOH(n))=96(10-29+27n^{2}).$

###### Proof

Let $G$ be the Dominating Planar Octahedral network $DPOH(n)$. By using edge
partition from Table 5, the result follows. From (4) we have

$M_{1}(DPOH(n))=\sum_{uv\in E(G)}(deg(u)+deg(v))=\sum_{uv\in
E_{j}(G)}\sum_{j=1}^{3}(deg(u)+deg(v)).$
$M_{1}(DPOH(n))=8|E_{1}(DPOH(n))|+12|E_{2}(DPOH(n))|+16|E_{3}(DPOH(n))|$

By doing some calculation, we get

$\Longrightarrow M_{1}(DPOH(n))=96(10-29+27n^{2}).$

∎Now, we compute $ABC$, $GA$, $ABC_{4}$ and $GA_{5}$ indices of Dominating
Planar Octahedral network $DPOH(n)$.

###### Theorem 2.2.6

Let $G\cong DPOH(n)$ be the network, then

$ABC(G)$ =
$\frac{1}{8}(72\sqrt{5}(1-3n+3n^{2})+6\sqrt{14}(5-13n+9n^{2})+12\sqrt{6}(1-5n+9n^{2}))$,
for $n\geq 1$

$GA(G)$ = $12(3+2\sqrt{2})(1-3n+3n^{2})$, for $n\geq 1$

$ABC_{4}(G)$ = $\frac{1}{440}(66\sqrt{78}+5280(-1+n)+120\sqrt{330}(-1+n)+\\\
24\sqrt{3410}(-1+n)+264\sqrt{29}n+88\sqrt{930}n+\\\
528\sqrt{35}(-1+2n)+132\sqrt{38}(-1+2n)+\\\
330\sqrt{46}(2-5n+3n^{2})+55\sqrt{94}(10-19n+9n^{2})+\\\
220\sqrt{35}(8-17n+9n^{2}))$, for $n\geq 3$

$GA_{5}(G)$
=$96+\frac{96}{23}\sqrt{33}(-1+n)+\frac{3}{2}\sqrt{55}(-1+n)+\frac{96}{17}\sqrt{66}(-1+n)+\\\
\frac{8}{7}\sqrt{110}(-1+n)-192n+8\sqrt{2}n+108n^{2}+\
\frac{48}{11}\sqrt{30}(-1+2n)+8\sqrt{2}(8-17n+9n^{2})$, for $n\geq 3$

###### Proof

By using edge partition given in Table 5 , we get the result. From $(5)$ it
follows that

$ABC(G)=\sum_{uv\in E(G)}\sqrt{\frac{deg(u)+deg(v)-2}{deg(u)\cdot
deg(v)}}=\sum_{uv\in
E_{j}(G)}\sum_{j=1}^{3}\sqrt{\frac{deg(u)+deg(v)-2}{deg(u)\cdot deg(v)}}.$
$ABC(G)=\frac{\sqrt{6}}{4}|E_{1}(DPOH(n))|+\frac{\sqrt{5}}{4}|E_{2}(DPOH(n))|+\frac{\sqrt{14}}{8}|E_{3}(RTSL(n))|$

By doing some calculation, we get

$\Longrightarrow
ABC(G)=\frac{1}{8}(72\sqrt{5}(1-3n+3n^{2})+6\sqrt{14}(5-13n+9n^{2})+12\sqrt{6}(1-5n+9n^{2}))$

from (6) we get

$GA(G)=\sum_{uv\in
E(G)}\frac{2\sqrt{deg(u)deg(v)}}{(deg(u)+deg(v))}=\sum_{uv\in
E_{j}(G)}\sum_{j=1}^{3}\frac{2\sqrt{deg(u)deg(v)}}{(deg(u)+deg(v))}.$

By doing some calculation, we get

$GA(G)=|E_{1}(DPOH(n))|+\frac{2\sqrt{2}}{3}|E_{2}(DPOH(n))|+|E_{3}(DPOH(n))|$
$\Longrightarrow GA=12(3+2\sqrt{2})(1-3n+3n^{2}).$

Let us consider an edge partition based on degree sum of neighbors of end
vertices. Then the edge set $E(DPOH(n))$ can be divided into twelve edge
partitions $E_{j}(DPOH(n)),4\leq j\leq 15$, where the edge partition
$E_{4}(DPOH(n))$ contains $12n-6$ edges $uv$ with $S_{u}=S_{v}=20$, the edge
partition $E_{5}(DPOH(n))$ contains $48n-24$ edges $uv$ with $S_{u}=20$ and
$S_{v}=24$, the edge partition $E_{6}(DPOH(n))$ contains $12n$ edges $uv$ with
$S_{u}=20$ and $S_{v}=40$, the edge partition $E_{7}(DPOH(n))$ contains
$12n-12$ edges $uv$ with $S_{u}=20$ and $S_{v}=44$, the edge partition
$E_{8}(DPOH(n))$ contains $54n^{2}-90n+36$ edges $uv$ with $S_{u}=24$ and
$S_{v}=24$, the edge partition $E_{9}(DPOH(n))$ contains $24n$ edges $uv$ with
$S_{u}=24$ and $S_{v}=40$, the edge partition $E_{10}(DPOH(n))$ contains
$48n-48$ edges $uv$ with $S_{u}=24$ and $S_{v}=44$, the edge partition
$E_{11}(DPOH(n))$ contains $108n^{2}-204n+96$ edges $uv$ with $S_{u}=24$ and
$S_{v}=48$, the edge partition $E_{12}(DPOH(n))$ contains $6$ edges $uv$ with
$S_{u}=40$ and $S_{v}=40$, the edge partition $E_{13}(DPOH(n))$ contains
$12n-12$ edges $uv$ with $S_{u}=40andS_{v}=44$ and the edge partition
$E_{14}(DPOH(n))$ contains $24n-24$ edges $uv$ with $S_{u}=44$ and $S_{v}=48$
and the edge partition $E_{15}(DPOH(n)$ contains $54n^{2}-114n+60$ edges $uv$
with $S_{u}=S_{v}=48.\\\ From(7)weget,\\\ $$ABC_{4}(G)=\sum_{uv\in
E(G)}\sqrt{\frac{S_{u}+S_{v}-2}{S_{u}S_{v}}}=\sum_{j=4}^{16}\sum_{uv\in
E_{j}(G)}\sqrt{\frac{S_{u}+S_{v}-2}{S_{u}S_{v}}}$$ABC_{4}(G)$=$\frac{\sqrt{38}}{20}|E_{4}(DPOH(n))|+\frac{\sqrt{35}}{20}|E_{5}(DPOH(n))|+\frac{\sqrt{29}}{20}|E_{6}(DPOH(n))|+\\\
0.2654|E_{7}(DPOH(n))|+\frac{\sqrt{46}}{4}|E_{8}(DPOH(n))|+0.2541|E_{9}(DPOH(n))|+\frac{1}{4}|E_{10}(DPOH(n))|+\frac{\sqrt{35}}{24}|E_{11}(DPOH(n))|+0.2518|E_{12}(DPOH(n))|+\frac{\sqrt{86}}{44}|E_{13}(DPOH(n))|+\frac{\sqrt{330}}{88}|E_{14}(DPOH(n))|+\frac{\sqrt{94}}{48}|E_{15}(DPOH(n))|$
$\Longrightarrow ABC_{4}$ =
$\frac{1}{440}(66\sqrt{78}+5280(-1+n)+120\sqrt{330}(-1+n)+\\\
24\sqrt{3410}(-1+n)+264\sqrt{29}n+88\sqrt{930}n+\\\
528\sqrt{35}(-1+2n)+132\sqrt{38}(-1+2n)+\\\
330\sqrt{46}(2-5n+3n^{2})+55\sqrt{94}(10-19n+9n^{2})+\\\
220\sqrt{35}(8-17n+9n^{2}))$
and from (8) we get

$GA_{5}(G)=\sum_{uv\in
E(G)}\frac{2\sqrt{S_{u}S_{v}}}{(S_{u}+S_{v})}=\sum_{j=4}^{15}\sum_{uv\in
E_{j}(G)}\frac{2\sqrt{S_{u}S_{v}}}{(S_{u}+S_{v})}$

$GA_{5}(G)=|E_{4}(DPOH(n))|+\frac{2\sqrt{30}}{14}|E_{5}D(POH(n))|+\frac{2\sqrt{2}}{3}|E_{6}(DPOH(n))|+\\\
\frac{\sqrt{55}}{8}|E_{7}(DPOH(n))|+|E_{8}(DPOH(n))|+\frac{\sqrt{15}}{4}|E_{9}(DPOH(n))|+\frac{2\sqrt{66}}{7}|E_{10}(DPOH(n))|+\frac{2\sqrt{2}}{3}|E_{11}(DPOH(n))|+\frac{2\sqrt{110}}{21}|E_{12}(DPOH(n))|+|E_{13}(DPOH(n))|+\frac{4\sqrt{33}}{23}|E_{14}(DPOH(n))|+|E_{1}5(DPOH(n))|$
$\Longrightarrow GA_{5}$ =
$96+\frac{96}{23}\sqrt{33}(-1+n)+\frac{3}{2}\sqrt{55}(-1+n)+\frac{96}{17}\sqrt{66}(-1+n)+\\\
\frac{8}{7}\sqrt{110}(-1+n)-192n+8\sqrt{2}n+108n^{2}+\\\
\frac{48}{11}\sqrt{30}(-1+2n)+8\sqrt{2}(8-17n+9n^{2})$.

The Comparison graphs of $ABC$, $GA$, $ABC_{4}$ and $GA_{5}$ for Planar
Octahedron network $POH(n)$ of dimension $n$, Triangular Prism network $TP(n)$
of dimension $n$ and Dominating Planar Octahedral network $DPOH(n)$ of
dimension $n$ are shown in fig. $2$, fig. $4$ and fig. $6$ respectively.

$(d_{u},d_{v})$ where $uv\in E(G)$ | Number of edges
---|---
$(4,4)$ | $18n^{2}+12n$
$(4,8)$ | $36n^{2}$
$(8,8)$ | $18n^{2}-12n$
Table 1: Edge partition of Planar Octahedron network $POH(n)$ based on degrees of end vertices of each edge. $(S_{u},S_{v})$ where $uv\in E(G)$ | Number of edges
---|---
$(20,20)$ | $6n$
$(20,24)$ | $24n$
$(20,40)$ | $12$
$(20,44)$ | $12n-12$
$(24,24)$ | $18n^{2}-18n$
$(24,40)$ | $24$
$(24,44)$ | $48n-48$
$(24,48)$ | $36n^{{}^{2}}-60n+24$
$(40,44)$ | $12$
$(44,44)$ | $12n-18$
$(44,48)$ | $12n-12$
$(48,48)$ | $18n^{2}-36n+18$
Table 2: Edge partition of David Derived network $DD(n)$ based on degrees of end vertices of each edge. Figure 4: Comparison of $ABC$, $GA$, $ABC_{4}$ and $GA_{5}$ for $DD(n)$ $(d_{u},d_{v})$ where $uv\in E(G)$ | Number of edges
---|---
$(3,3)$ | $18n^{2}+6n$
$(3,6)$ | $18n^{2}+6n$
$(6,6)$ | $18n^{2}-12n$
Table 3: Edge partition of Triangular Prism network $TP(n)$ based on degrees of end vertices of each edge. $(S_{u},S_{v})$ where $uv\in E(G)$ | Number of edges
---|---
$(9,12)$ | $12n$
$(9,15)$ | $6n$
$(12,12)$ | $8n^{2}-12$
$(12,24)$ | $12$
$(12,27)$ | $24n-24$
$(12,30)$ | $18n^{2}-30n+12$
$(15,24)$ | $12$
$(15,27)$ | $12n-12$
$(24,27)$ | $12$
$(27,27)$ | $12n-18$
$(27,30)$ | $12n-12$
$(30,30)$ | $18n^{2}-36n+18$
Table 4: Edge partition of Triangular Prism network $TP(n)$ based on degrees of end vertices of each edge. $(d_{u},d_{v})$ where $uv\in E(G)$ | Number of edges
---|---
$(4,4)$ | $54n^{2}-30n+6$
$(4,8)$ | $108n^{2}-108n+36$
$(8,8)$ | $54n^{2}-78n+30$
Table 5: Edge partition of Dominating Planar Octahedral network $DPOH(n)$ based on degrees of end vertices of each edge. $(S_{u},S_{v})$ where $uv\in E(G)$ | Number of edges
---|---
$(20,20)$ | $12n-6$
$(20,24)$ | $48n-24$
$(20,40)$ | $12n$
$(20,44)$ | $12n-12$
$(24,24)$ | $54n^{2}-90+36n$
$(24,40)$ | $24n$
$(24,44)$ | $48n-48$
$(24,48)$ | $108n^{{}^{2}}-204n+96$
$(40,40)$ | $6$
$(40,44)$ | $12n-12$
$(44,48)$ | $24n-24$
$(48,48)$ | $54n^{2}-114n+60$
Table 6: Edge partition of Dominating Planar Octahedral network $DPOH(n)$
based on degrees of end vertices of each edge.

The graphical representations of topological indices of these networks are
depicted in Figure 5 and Figure 6 for certain values of $m$. By varying the
different values of $m$, the graphs are increasing. These graphs shows the
correctness of results.
Conclusion:

In this paper, certain degree based topological indices, to be specific the
Randi$\acute{c}$, Zagreb, $ABC$, $GA$, $ABC_{4}$ and $GA_{5}$ indices for
planar octahedron networks just because and scientific shut equations for
these systems were resolved which will help the general population working in
system science to comprehend and investigate the basic topologies of these
systems. Later on, we are intrigued to structure some new models/systems and
after that review their topological records which will be very useful to
comprehend their fundamental topologies.

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# Neutrino oscillations in gravitational fields and astrophysical applications

Maxim Dvornikov  Pushkov Institute of Terrestrial Magnetism, Ionosphere and
Radiowave Propagation (IZMIRAN), 108840 Moscow, Troitsk, Russia
<EMAIL_ADDRESS>
###### Abstract

The neutrino propagation and oscillations in various gravitational fields are
studied. First, we consider the neutrino scattering off a black hole
accounting for the neutrino spin precession. Then, we study the evolution of
flavor neutrinos in stochastic gravitational waves. The astrophysical
applications of the obtained results are considered.

††preprint: MUPB/Conference section:

Oscillations of neutrinos, confirmed experimentally, allow one to explore
physics beyond the standard model. This phenomenon can take place only for
nonzero masses and mixing between different neutrino generations. We also
mention the possibility of transitions between active left handed and sterile
right handed neutrinos. This process is called neutrino spin oscillations.

The interaction with external fields can modify the process of neutrino
oscillations. Gravitational fields, in spite of their weakness, can also
affect neutrino oscillations. We studied neutrino spin oscillations in
background matter under the influence of an electromagnetic field in static
curved spacetimes in Refs. [1, 2]. Spin oscillations of neutrinos in their
scattering off a black hole (BH) were analyzed in Refs. [3, 4, 5]. This
research was inspired by the recent observation of the event horizon
silhouette of a supermassive BH (SMBH) in the center of M87. An accretion disk
around BH can be a source of both photons and neutrinos. Neutrino spin
oscillations in a gravitational wave (GW) were considered in Ref. [6]. We
examined neutrino flavor oscillations in stochastic GWs in Refs. [7, 8, 9].
The study of the neutrino interaction with GWs was motivated by the direct
detection of GWs in terrestrial experiments. In the present work, we summarize
our results in Refs. [5, 9].

First, we study neutrino spin oscillations in the particle scattering off a
rotating BH surrounded by a dense magnetized accretion disk. The invariant
neutrino spin vector $\bm{\zeta}$, which is defined in the rest frame of the
locally Minkowskian frame, obeys the precession equation [2, 5],
$\dot{\bm{\zeta}}=2(\bm{\zeta}\times\bm{\Omega})$. The components of the
vector $\bm{\Omega}$ incorporate the contributions of all the external fields.
They are given in the explicit form in Ref. [5]. Note that one has to account
for both the neutrino spin precession and the evolution of the neutrino
velocity to describe the neutrino polarization. We suppose that neutrinos move
along the geodesic lines in their scattering off BH.

We use the Kerr metric for the description of the spacetime around a rotating
SMBH with $M\sim 10^{8}\,M_{\odot}$, which is surrounded by a relativistic
accretion disk. The electron number density in the disk scales as
$n_{e}(r)\propto n_{e}^{(0)}r^{-3/2}$, where $n_{e}^{(0)}\sim
10^{18}\,\text{cm}^{-3}$ is the number density in the vicinity of SMBH. We
suppose that the magnetic field in the disk is poloidal and its strength
scales as $B(r)\propto B_{0}r^{-5/4}$. The magnetic field in the vicinity of
BH is taken to be $B_{0}=3.2\times 10^{2}\,\text{G}$. Neutrinos are supposed
to be Dirac particles having the anomalous magnetic moment
$\mu=10^{-14}\mu_{\mathrm{B}}$, where $\mu_{\mathrm{B}}$ is the Bohr magneton.
The matter of the accretion disk is taken to rotate around SMBH on circular
orbits, which are the realistic geodesics for plasma particles forming this
disk. Note that an accretion disk can both corotate and counter-rotate BH. We
suppose that neutrinos move in the equatorial plane only.

The total neutrino fluxes scattered off BH accounting for the spin evolution
for different angular momenta of BH are shown in Fig. 1. One can see in Fig. 1
that there are spikes in the fluxes at certain scattering angles. The
appearance of these spikes can be accounted for qualitatively by the neutrino
interaction with moving matter. In the neutrino gravitational scattering,
there are situations when the neutrino velocity has a nonzero angle with
respect to the plasma velocity. In this case, a neutrino spin-flip can happen.

Figure 1: (a) The total flux $F_{\nu,d}^{(\text{tot})}$, normalized by the
flux of scalar particles $F_{0,d}$, for the direct scattering (a detector is
bent towards the BH rotation) for BH with the angular momentum $J=M^{2}$
versus the scattering angle $\chi$. (b) The total flux
$F_{\nu,r}^{(\text{tot})}$ for the retrograde scattering (a detector is bent
in the opposite direction with respect to the BH rotation) for BH with
$J=0.2M^{2}$. The accretion disk counter-rotates BH in both panels. These
figures are taken from Ref. [5].

One can see in Fig. 1 that the observed neutrino flux can be suppressed by
several percent because of spin oscillations in the neutrino gravitational
scattering. The major reduction of the flux happens for the backward neutrino
scattering when $\chi=\pi$. We hope that the predicted effect can be observed
by current or future neutrino telescopes.

Now, we turn to the problem of the propagation and flavor oscillations of
supernova (SN) neutrinos under the influence of stochastic GWs. The flavor
content of a neutrino beam can be described by the $3\times 3$ density matrix
$\rho_{f}$ in the flavor basis. We can define the new density matrix
$\rho^{\prime}=\exp(\mathrm{i}H_{m}^{(\mathrm{vac})}t)U^{\dagger}\rho_{f}U\exp(-\mathrm{i}H_{m}^{(\mathrm{vac})}t)$,
where $(H_{m}^{(\mathrm{vac})})_{ab}=\tfrac{m_{a}^{2}}{2E}\delta_{ab}$ is the
vacuum oscillations Hamiltonian in the mass basis corresponding to the masses
$m_{a}$, $E$ is the mean neutrino energy, and $U$ is the mixing matrix. In the
presence of stochastic GWs, the averaged density matrix
$\langle\rho^{\prime}\rangle$ was shown in Ref. [9] to obey the equation
$\tfrac{\mathrm{d}}{\mathrm{d}t}\left\langle\rho^{\prime}\right\rangle(t)=g(t)[H_{m}^{(\mathrm{vac})},[H_{m}^{(\mathrm{vac})},\left\langle\rho^{\prime}\right\rangle(t)]]$,
where the function $g(t)$ contains the correlators of the amplitudes
$h_{+,\times}$ of the ‘plus’ and ‘cross’ polarizations of GW.

We take that a beam of neutrinos is emitted in a core-collapsing SN, which can
be considered as almost a point-like source. In this case, only solar
neutrinos oscillations channel has a nonvanishing contribution to the observed
fluxes. We also suppose that these SN neutrinos interact with stochastic GWs
emitted by merging SMBHs. The equation for $\langle\rho^{\prime}\rangle$ can
be solved analytically in this situation. The behavior of the GW contribution
to the flux of electron neutrinos is shown in Fig. 2. In Fig. 2, we
demonstrate the evolution of $\Delta
F_{\nu_{e}}=F_{\nu_{e}}-F^{(\text{vac})}_{\nu_{e}}$ along the neutrino
propagation distance $x=\tau L$. Here $F_{\nu_{e}}$ is the flux accounting for
the GW contribution and $F^{(\text{vac})}_{\nu_{e}}$ is the flux corresponding
to vacuum oscillations. The fluxes of $\nu_{\mu}$ and $\nu_{\tau}$ are
considered in Ref. [9].

Figure 2: (a) The correction the electron neutrino flux $\Delta
F_{\nu_{e}}\propto\Delta P_{\nu_{e}}$, where $P_{\nu_{e}}$ is the probability
to detect $\nu_{e}$, owing to the neutrino interaction with stochastic GWs.
The parameters of the system are $\Delta m_{21}^{2}=7.5\times
10^{-5}\,\text{eV}^{2}$, $E=10\,\text{MeV}$, $\theta_{12}=0.6$,
$\theta_{23}=0.85$, $\theta_{13}=0.15$, $\delta_{\mathrm{CP}}=3.77$,
$L=10\,\text{kpc}$ is the maximal propagation distance, $\Omega_{0}=10^{-9}$
is the maximal normalized spectral energy density of GWs, and
$f_{\mathrm{min}}=10^{-10}\,\text{Hz}$ is the minimal frequency of the GW
spectrum. The normal neutrino mass ordering is adopted. This figure is taken
from Ref. [9].

One can see in Fig. 2 that the interaction of SN neutrinos with stochastic GWs
can modify the observed flux, e.g., of electron neutrinos by a few percent. It
means that one expects the change of the SN neutrinos fluxes by $\sim\pm 350$
events, in case of the Super-Kamiokande, and by $\sim\pm 3750$ events, for the
Hyper-Kamiokande if a SN explosion happens in our Galaxy.

## References

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