Computational analysis of NM-polynomial based topological indices and graph-
entropies of carbon nanotube Y-junctions

Sohan Lal1, Vijay Kumar Bhat1,∗, Sahil Sharma1

1School of Mathematics, Shri Mata Vaishno Devi University,

Katra-$182320$, Jammu and Kashmir, India.

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Carbon nanotube Y-junctions are of great interest to the next generation of
innovative multi-terminal nanodevices. Topological indices are graph-
theoretically based parameters that describe various structural properties of
a chemical molecule. The entropy of a graph is a topological descriptor that
serves to characterize the complexity of the underlying molecular graph. The
concept of entropy is a physical property of a thermodynamic system. Graph
entropies are the essential thermophysical quantities defined for various
graph invariants and are applied to measure the heterogeneity and relative
stabilities of molecules. In this paper, several neighborhood degree sum-based
topological indices including graph-based entropies of carbon nanotube
Y-junction graphs are computed.

Keywords: Armchair carbon nanotube, graph entropy, NM-polynomial, topological
indices, Y-junction graph.

MSC (2020): 05C10, 05C35, 05C90

##  1 Introduction

Nanotechnology is currently popular because of its evolving, electron transfer
property and low-cost implementation. Nanotubes [1], were discovered in $1985$
and carbon nanotubes [2] in $1991$. In nanoscience and technology, branched or
non-straight carbon nanotubes such as L, T, X, and Y have a lot of
applications in electronic devices, such as three-terminal transistors, multi-
terminal nanoelectronics, switches, amplifiers, etc., [3, 4, 5, 6, 7, 8].
These junctions are a great option for the production of nanoscale electronic
devices with better switching and reliable transport properties at room
temperature. For more applications of carbon nanotube Y-junctions, we refer to
[9, 10, 11].

The first proposed branched carbon nanotube was of Y shape, commonly known as
Y-junction or three-terminal junction. These junctions are classified as an
armchair, zig-zag, or chiral depending on the chirality of connected carbon
nanotubes. Also, they can be single-walled or multi-walled, symmetric or
asymmetric, capped or uncapped. A carbon nanotube is called uncapped if both
ends are open. A Y-junction is called symmetric if the nanotubes joining in
the Y shape are identical, heptagons appeared isolated, and are distributed
symmetrically. For various symmetric and asymmetric carbon nanotube
Y-junctions, we refer to [12, 13, 14, 15].

A carbon nanotube Y-junction is formed by joining three identical carbon
nanotubes in a Y-shaped pattern. These junctions contain exactly six hexagons
as well as heptagons at the branching points. The first structural model of
symmetrical single-walled armchair carbon nanotube Y-junctions was proposed by
Chernozatonskii [16] and Scuseria [17], independently, in $1992$. These
junctions were experimentally observed [18] in $1995$. For more applications
and properties of carbon nanotube Y-junction graphs, we refer to [19, 20, 21].

Mathematical chemistry is a branch of theoretical chemistry that employs
mathematical techniques to explain the molecular structure of a chemical
molecule and its physicochemical properties. Molecular graphs are a visual
representation of a chemical molecule with vertices representing atoms and
edges representing bonds between the atoms [22]. Let $G=(V(G),E(G))$ be a
molecular graph with vertex set $V(G)$ and edge set $E(G)$. The order of a
molecular graph $G$ is defined as the total number of vertices in $G$, denoted
by $\lvert V(G)\rvert$, and the number of edges in $G$ is called size of $G$,
denoted by $\lvert E(G)\rvert$. Any edge of the graph connecting its vertices
$u$ and $v$, is denoted by $e=uv\in E(G)$. Two vertices of graph $G$ are said
to be adjacent if there exists an edge between them. The degree of vertex
$v\in V(G)$, denoted by $d(v)$, is defined as the number of vertices that are
adjacent to vertex $v$, i.e., $d(v)$= $\lvert\\{u:e=uv\in E(G)\\}\rvert$. The
neighborhood degree sum of vertex $v\in V(G)$ is denoted by $d_{n}(v)$, and is
defined as the sum of the degrees of all vertices that are adjacent to $v$,
i.e., $d_{n}(v)=\sum\limits_{u}d(v)$: $uv\in E(G)$. The minimum cardinality of
the set $K\subseteq V(G)$ such that $G\setminus K$ is disconnected graph is
called connectivity or vertex-connectivity of a connected graph $G$. The
connected graph $G$ is said to be $k$-connected if its connectivity is $k$.

Topological indices are the numerical values calculated from molecular graphs
to describe various structural properties of the chemical molecule. They are
frequently used to model many physicochemical properties in various
quantitative structure-property/activity relationship (QSPR/QSAR) studies [23,
24, 25]. In $1947$, the chemist Harold Wiener [26] initiated the concept of
topological indices. Since then, various topological indices have been
introduced, and a lot of research has been conducted toward computing the
indices for different molecular graphs and networks. A topological index based
on the degree of end vertices of an edge can predict various physicochemical
properties of the molecule, such as heat of formation, strain energy, entropy,
enthalpy, boiling points, flash point, etc., without using any weight lab
[24].

The Zagreb indices and their variations have been used to investigate
molecular complexity, ZE-isomerism, and chirality [27]. In general, the Zagreb
indices have shown applicability for deriving multilinear regression models.
Ghorbani and Hosseinzadeh [28] introduced the third version of the Zagreb
index and shows that this index shows a good correlation with acentric factor
and entropy of the octane isomers. Mondal et al. [29] introduced neighborhood
degree sum-based topological indices namely neighborhood version of forgotten
topological index and neighborhood version of second modified Zagreb index and
discuss some mathematical properties and degeneracy of these novel indices.
For more neighborhood degree sum-based topological indices, their properties,
and applications, we refer to [24, 30, 31].

The process of computing the topological indices of a molecular graph from
their definitions is complex and time-consuming. Thus, for a particular family
of graphs and networks, algebraic polynomials play an important role in
reducing the computational time and complexity when computing its topological
indices. In short, with the help of algebraic polynomials, one can easily
compute various kinds of graph indices within a short span of time. The NM-
polynomial plays vital role in the computation of neighborhood degree sum-
based topological indices. Let $d_{n}(v)$ denotes the neighborhood degrees sum
of vertex $v\in V(G)$. Then, the neighborhood M-polynomial (NM-polynomial) of
$G$ is defined as [30, 32, 33]

$\displaystyle NM(G;x,y)=\sum\limits_{i\leq j}\lvert E_{ij}(G)\rvert
x^{i}y^{j}$ (1)

where, $\lvert E_{ij}(G)\rvert$, $i,j\geq 1$, be the number of all edges
$e=uv\in E(G)$ such that $\\{d_{n}(u)=i,d_{n}(v)=j\\}$.

Recently, various neighborhood degree sum-based topological indices have been
computed via the NM-polynomial technique. For example, Mondal et al. [30, 34]
obtained some neighborhood and multiplicative neighborhood degree sum-based
indices of molecular graphs by using their NM-polynomials. Kirmani et al. [24]
and Mondal et al. [35], investigated some neighborhood degree sum-based
topological indices of antiviral drugs used for the treatment of COVID-$19$
via the NM-polynomial technique. Shanmukha et al. [36] computed the
topological indices of porous graphene via NM-polynomial method. For more
neighborhood degree sum-based topological indices via NM-polynomials, we refer
to [24, 35, 37, 38].

Some neighborhood degree sum-based topological indices and their derivation
from NM-polynomial are given in Table $1$.

Table 1: Description of some topological indices and its derivation from NM-
polynomial

Topological index | Formula |  Derivation from $NM(G;x,y)$
---|---|---
Third version of Zagreb index [28]: $NM_{1}(G)$ | $\sum\limits_{uv\in E(G)}\big{(}d_{n}(u)+d_{n}(v)\big{)}$ | $(D_{x}+D_{y})(NM(G;x,y))\rvert_{x=y=1}$
Neighborhood second Zagreb index [29]: $NM_{2}(G)$ | $\sum\limits_{uv\in E(G)}\big{(}d_{n}(u)d_{n}(v)\big{)}$ | $(D_{x}D_{y})(NM(G;x,y))\rvert_{x=y=1}$
Neighborhood second modified Zagreb index [30]: ${}^{nm}M_{2}(G)$ | $\sum\limits_{uv\in E(G)}\big{(}\frac{1}{d_{n}(u)d_{n}(v)}\big{)}$ | $(S_{x}S_{y})(NM(G;x,y))\rvert_{x=y=1}$
Neighborhood forgotten topological index [29]: $NF(G)$ | $\sum\limits_{uv\in E(G)}\big{(}d_{n}^{2}(u)+d_{n}^{2}(v)\big{)}$ | $(D^{2}_{x}+D^{2}_{y})(NM(G;x,y))\rvert_{x=y=1}$
Third NDe index [30]: $ND_{3}(G)$ | $\sum\limits_{uv\in E(G)}d_{n}(u)d_{n}(v)(d_{n}(u)+d_{n}(v))$ | $D_{x}D_{y}(D_{x}+D_{y})(NM(G;x,y))\rvert_{x=y=1}$
Neighborhood general Randic index [30]: $NR_{\alpha}(G)$ | $\sum\limits_{uv\in E(G)}d_{n}^{\alpha}(u)d_{n}^{\alpha}(v)$ | $(D^{\alpha}_{x}D^{\alpha}_{y})(NM(G;x,y))\rvert_{x=y=1}$
Neighborhood inverse Randic index [30]: $NRR_{\alpha}(G)$ | $\sum\limits_{uv\in E(G)}\frac{1}{d_{n}^{\alpha}(u)d_{n}^{\alpha}(v)}$ | $(S^{\alpha}_{x}S^{\alpha}_{y})(NM(G;x,y))\rvert_{x=y=1}$
Fifth NDe index [30]: $ND_{5}(G)$ | $\sum\limits_{uv\in E(G)}\big{(}\frac{d_{n}^{2}(u)+d_{n}^{2}(v)}{d_{n}(u)d_{n}(v)}\big{)}$ | $(D_{x}S_{y}+S_{x}D_{y})(NM(G;x,y))\rvert_{x=y=1}$
Neighborhood harmonic index [30]: $NH(G)$ | $\sum\limits_{ab\in E(G)}\frac{2}{d_{n}(u)+d_{n}(v)}$ | $2S_{x}T(NM(G;x,y))\rvert_{x=y=1}$
Neighborhood inverse sum indeg index [30]: $NI(G)$ | $\sum\limits_{uv\in E(G)}\big{(}\frac{d_{n}(u)d_{n}(v)}{d_{n}(u)+d_{n}(v)}\big{)}$ | $(S_{x}TD_{x}D_{y})(NM(G;x,y))\rvert_{x=y=1}$

where, $D_{x}=x\big{(}\frac{(\partial(NM(G;x,y))}{\partial x}\big{)}$,
$D_{y}=y\big{(}\frac{(\partial(NM(G;x,y))}{\partial y}\big{)}$,
$S_{x}=\int_{0}^{x}\frac{NM(G;t,y)}{t}dt$,
$S_{y}=\int_{0}^{y}\frac{NM(G;x,t)}{t}dt$, $T(NM(G;x,y))=NM(G;x,x)$.

In chemical graph theory, the determination of the structural information
content [39] of a graph is mostly based on the vertex partition of a graph to
obtain a probability distribution of its vertex set [40]. Based on such a
probability distribution, the entropy of a graph can be defined. Thus, the
structural information content of a graph is defined as the entropy of the
underlying graph topology. The concept of graph entropy or entropy of graph
was first time appeared in [41], where molecular graphs are used to study the
information content of an organism. Entropy-based methods are powerful tools
to investigate various problems in cybernetics, mathematical chemistry,
pattern recognition, and computational physics [22, 39, 42, 43, 44].

Entropy is a measure of randomness, uncertainty, heterogeneity, or lack of
information in a system. Based on information indices, there are various
approaches to deriving graph entropy from the topological structure of a given
chemical molecule [45]. For example, Trucco [39] and Rashevsky [41] defined
graph entropies in terms of degree of vertex, extended degree sequences, and
number of vertices of a molecular graph. Tan and Wu [46] study network
heterogeneity by using vertex-degree based entropies. Mowshowitz defined the
entropy of a graph in terms of equivalence relations defined on the vertex set
of a graph and discussed some properties related to structural information
[47, 48, 49, 50].

Recently, Shabbir and Nadeem [51] defined graph entropies in terms of
topological indices for the molecular graphs of carbon nanotube Y-junctions
and developed the regression models between the graph entropies and
topological indices. Nadeem et al. [52] calculated some degree-based
topological indices for armchair carbon semicapped and capped nanotubes and
investigated their chemical and physical properties. Bača et al. [53] computed
some degree-based topological indices of a carbon nanotube network and studied
its properties. Azeem et al. [54] calculated some M-polynomials based
topological indices of carbon nanotube Y-junctions and their variants. Ahmad
[55], studied some ve-degree based topological indices of carbon nanotube
Y-junctions and discussed their properties. Ayesha [56] calculated the bond
energy of symmetrical single-walled armchair carbon nanotube Y-junctions and
developed regression models between bond energy and topological indices. Rahul
et al. [57] calculated some degree-based topological indices and graph-
entropies of graphene, graphyne, and graphdiyne by using Shannon’s approach.

The above-mentioned literature and applications of carbon nanotubes in the
field of nanoscience and technology inspired us to develop more research on
the molecular structure of carbon nanotube Y-junction and their variants. In
addition, no work has been reported on NM-polynomial based topological indices
and index-entropies of Y-junction graphs. Therefore, the main contribution of
this study includes the following:

  * •

Computation of NM-polynomials of carbon nanotube Y-junction graphs.

  * •

Computation of some neighborhood degree sum-based topological indices from NM-
polynomials.

  * •

Some graph index-entropies in terms of topological indices are defined and
computed.

  * •

Comparative analysis of obtained topological indices and graph index-entropies
of Y-junction graphs.

##  2 Aim and Methodology

We use the edge partition technique, graph-theoretical tools, combinatorial
computation, and the degree counting method to derive our results. The degree
of end vertices is used to generate the patterns of edge partitions of the
Y-junction graphs. Using such partitions, a general expression of NM-
polynomials is derived. Then, several neighborhood degree sum-based
topological indices are obtained from the expression of these NM-polynomials
with the help of Table $1$. Also, graph index- entropies in terms of
topological indices have been defined by using edge-weight functions and
computed for Y-junction graphs.

The paper is structured as follows: In Section $3$, we define topological
index-based graph entropies. The Y-junction graphs and their constructions are
described in Section $4$. In Section $5$, the general expression of the NM-
polynomials and neighborhood degree sum-based topological indices of
Y-junction graphs are presented. Section $6$ describes the graph index-
entropies of Y-junction graphs. The numerical analysis of the findings is
discussed in Section $7$. Finally, the conclusion is drawn and discussed in
Section $8$.

##  3 Definitions and Preliminaries

In this section, we define graph index-entropies in terms of an edge-weight
function. In $2008$, Dehmer [40] defined the entropy for a connected graph $G$
as follows:

######  Definition 1.

[40] Let $G=(V(G),E(G))$ be a connected graph of order $n$ and $g$ be an
arbitrary information functional. Then the entropy of $G$ is defined as

$H_{g}(G)=-\sum\limits_{i=1}^{n}\frac{g(v_{i})}{\sum\limits_{i=1}^{n}g(v_{i})}log\biggl{(}\frac{g(v_{i})}{\sum\limits_{i=1}^{n}g(v_{i})}\biggr{)}.$
(2)

Since an information function defined on the vertex set of a graph is an
arbitrary function. Hence, Dehmer’s definition shows the possibility of
producing various graph entropies for a variation in the selection of
information functionals. For such graph entropy, we can refer to [58, 59, 60].

Let $\beta:E(G)\to\mathbb{R}^{+}\cup\\{0\\}$ be an edge-weight function and
$d_{n}(u)=\sum\limits_{uv\in E(G)}d(u)$, denotes the sum of degrees of end
vertices of an edges incident to vertex $u\in V(G)$ (also known as
neighborhood degree-sum of vertex $u$). Then, for eight different edge-weight
functions, the third-version of Zagreb index, neighborhood second Zagreb
index, neighborhood forgotten topological index, neighborhood second modified
Zagreb index, third NDe index, fifth NDe index, neighborhood harmonic index
and neighborhood inverse sum indeg index-entropies have been defined in the
following manner:

  * •

Third-version of Zagreb index-entropy: If $e=uv$ is an edge of a connected
graph $G$ and $\beta_{1}(e)=d_{n}(u)+d_{n}(v)$ is an edge-weight function
defined on $E(G)$. Then, the third-version of Zagreb index is

$NM_{1}(G)=\sum\limits_{e=uv\in E(G)}\beta_{1}(e)=\sum\limits_{e=uv\in
E(G)}d_{n}(u)+d_{n}(v).$ (3)

Equation (2) for this edge-weight function gives us

$\displaystyle H_{\beta_{1}}(G)$ $\displaystyle=$
$\displaystyle-\sum\limits_{e\in E(G)}\frac{\beta_{1}(e)}{\sum\limits_{e\in
E(G)}\beta_{1}(e)}log\bigg{(}\frac{\beta_{1}(e)}{\sum\limits_{e\in
E(G)}\beta_{1}(e)}\bigg{)}$ $\displaystyle=$
$\displaystyle-\frac{1}{\sum\limits_{e\in E(G)}\beta_{1}(e)}\sum\limits_{e\in
E(G)}\beta_{1}(e)\bigg{(}log(\beta_{1}(e))-log\sum\limits_{e\in
E(G)}\beta_{1}(e)\bigg{)}$ $\displaystyle=$
$\displaystyle-\frac{1}{\sum\limits_{e\in E(G)}\beta_{1}(e)}\sum\limits_{e\in
E(G)}\beta_{1}(e)log(\beta_{1}(e))+\frac{1}{\sum\limits_{e\in
E(G)}\beta_{1}(e)}\sum\limits_{e\in
E(G)}\beta_{1}(e)log\bigg{(}\sum\limits_{e\in E(G)}\beta_{1}(e)\bigg{)}$
$\displaystyle=$ $\displaystyle log\bigg{(}\sum\limits_{e\in
E(G)}\beta_{1}(e)\bigg{)}-\frac{1}{\sum\limits_{e\in
E(G)}\beta_{1}(e)}\sum\limits_{e\in E(G)}\beta_{1}(e)log(\beta_{1}(e)).$

On replacing $\sum\limits_{e\in E(G)}\beta_{1}(e)$ by $NM_{1}(G)$ in the above
equation, we get the following third-version of Zagreb index-entropy

$H_{\beta_{1}}(G)=log(NM_{1}(G))-\frac{1}{NM_{1}(G)}\sum\limits_{e\in
E(G)}{\beta}_{1}(e)log\beta_{1}(e).$ (4)

Similarly, we define other graph index-entropies as follows:

  * •

Neighborhood second Zagreb index-entropy: For $\beta_{2}(e)=d_{n}(u)d_{n}(v)$,
the neighborhood second Zagreb index and neighborhood second Zagreb index-
entropy are

$NM_{2}(G)=\sum\limits_{e=uv\in E(G)}d_{n}(u)d_{n}(v),$ (5)

and

$H_{\beta_{2}}(G)=log(NM_{2}(G))-\frac{1}{NM_{2}(G)}\sum\limits_{e\in
E(G)}{\beta}_{2}(e)log\beta_{2}(e).$ (6)

  * •

Neighborhood forgotten topological index-entropy: For
$\beta_{3}(e)=d_{n}^{2}(u)+d_{n}^{2}(v)$, the neighborhood forgotten
topological index and neighborhood forgotten topological index-entropy are

$NF(G)=\sum\limits_{e=uv\in E(G)}d_{n}^{2}(u)+d_{n}^{2}(v),$ (7)

and

$H_{\beta_{3}}(G)=log(NF(G))-\frac{1}{NF(G)}\sum\limits_{e\in
E(G)}{\beta}_{3}(e)log\beta_{3}(e).$ (8)

  * •

Neighborhood second modified Zagreb index-entropy: For
$\beta_{4}(e)=\frac{1}{d_{n}(u)d_{n}(v)}$, the neighborhood second modified
Zagreb index and neighborhood second modified Zagreb index-entropy are

${}^{nm}M_{2}(G)=\sum\limits_{e=uv\in E(G)}\frac{1}{d_{n}(u)d_{n}(v)},$ (9)

and

$H_{\beta_{4}}(G)=log(^{nm}M_{2}(G))-\frac{1}{{}^{nm}M_{2}(G)}\sum\limits_{e\in
E(G)}{\beta}_{4}(e)log\beta_{4}(e).$ (10)

  * •

Third NDe index-entropy: For
$\beta_{5}(e)=d_{n}(u)d_{n}(v)\big{(}d_{n}(u)+d_{n}(v)\big{)}$, the third NDe
index and third NDe index-entropy are

$ND_{3}(G)=\sum\limits_{e=uv\in
E(G)}d_{n}(u)d_{n}(v)\big{(}d_{n}(u)+d_{n}(v)\big{)},$ (11)

and

$H_{\beta_{5}}(G)=log(ND_{3}(G))-\frac{1}{ND_{3}(G)}\sum\limits_{e\in
E(G)}{\beta}_{5}(e)log\beta_{5}(e).$ (12)

  * •

Fifth NDe index-entropy: For
$\beta_{6}(e)=\frac{d_{n}(u)}{d_{n}(v)}+\frac{d_{n}(v)}{d_{n}(u)}$, the fifth
NDe index and fifth NDe index-entropy are

$ND_{5}(G)=\sum\limits_{e=uv\in
E(G)}\frac{d_{n}(u)}{d_{n}(v)}+\frac{d_{n}(v)}{d_{n}(u)},$ (13)

and

$H_{\beta_{6}}(G)=log(ND_{5}(G))-\frac{1}{ND_{5}(G)}\sum\limits_{e\in
E(G)}{\beta}_{6}(e)log\beta_{6}(e).$ (14)

  * •

Neighborhood harmonic index-entropy: For
$\beta_{7}(e)=\frac{2}{d_{n}(u)+d_{n}(v)}$, the neighborhood harmonic index
and neighborhood harmonic index-entropy are

$NH(G)=\sum\limits_{e=uv\in E(G)}\frac{2}{d_{n}(u)+d_{n}(v)},$ (15)

and

$H_{\beta_{7}}(G)=log(NH(G))-\frac{1}{NH(G)}\sum\limits_{e\in
E(G)}{\beta}_{7}(e)log\beta_{7}(e).$ (16)

  * •

Neighborhood inverse sum indeg index-entropy: For
$\beta_{8}(e)=\frac{d_{n}(u)d_{n}(v)}{d_{n}(u)+d_{n}(v)}$, the neighborhood
inverse sum index and neighborhood inverse sum index-entropy are

$NI(G)=\sum\limits_{e=uv\in E(G)}\frac{d_{n}(u)d_{n}(v)}{d_{n}(u)+d_{n}(v)},$
(17)

and

$H_{\beta_{8}}(G)=log(NI(G))-\frac{1}{NI(G)}\sum\limits_{e\in
E(G)}{\beta}_{8}(e)log\beta_{8}(e).$ (18)

##  4 Y-Junction Graphs

The Y-junctions examined in this study are created by the covalent connection
of three identical single-walled carbon nanotubes crossing at an angle of
$120^{\circ}$ and are uniquely determined by their chiral vector
$v=nv_{1}+nv_{2}$, where $v_{1}$ and $v_{2}$ are graphene sheet lattice
vectors and $n$ is non-negative integer. Let $m\geq 1$ and $n\geq 4$ be an
even integer. Then, an uncapped symmetrical single-walled carbon nanotube
Y-junction is made up of an armchair $Y(n,n)$ and three identical single-
walled armchair carbon nanotubes $T_{m}(n,n)$ each of length $m$ (layers of
hexogones), denoted by $Y^{m}(n,n)$. In $Y^{m}(n,n)$, we have
$\frac{3}{4}n^{2}-\frac{3}{2}n+5$ faces including three openings (where the
tubes meet to the amchair) each of chirality $(n,n)$, six heptagones, and
$\frac{3}{4}n^{2}-\frac{3}{2}n-4$ hexagones. In addition, the tube
$T_{m}(n,n)$ contains $2mn$ hexagonal faces.

Let $n$, $m$, and $l$ be positive integers with $m\geq 1$ and $n=2l$, for some
$l\geq 2$. Then $J=J^{m}(n,n)$ be the $Y$-junction graph of $Y^{m}(n,n)$. It
has $9l^{2}-3l+2$ hexagonal rings along with six heptagons. The graph $J$ is
of order $6l^{2}+18l+6+24ml$ and size $9l^{2}+21l+9+36ml$. It has
$6l^{2}+12l+6+24ml$ vertices of degree three and $12l$ vertices of degree two.
Note that graph $J$ is a $2$-conneced graph.

Along with $2$-connected Y-junction graph $J$, the $1$-connected Y-junction
graphs have also been taken into consideration. These graphs are obtained by
adding pendants to the degree $2$ vertices of the $2$-connected graph $J$.
Note that, each tube of $J$ has $2n$ vertices of degree $2$. Therefore, the
graph $J$ has $6n$ vertices of degree $2$.

The graph obtained by connecting $2n$ pendants to any one tube in $J$ is
denoted by $J_{1}$, and we call it as second type Y-junction graph. The order
and size of graph $J_{1}$ are $6l^{2}+22l+6+24ml$ and $9l^{2}+25l+9+36ml$,
respectively. The graph $J_{2}$ represents a graph which is obtained by
attaching $4n$ pendants to any two tubes of $J$ and we call it as third type
Y-junction graph. In $J_{2}$, we have $6l^{2}+26l+6+24ml$ vertices and
$9l^{2}+29l+9+36ml$ edges. The graph obtained by joining $6n$ pendants to all
the three tubes of $J$ is denoted by $J_{3}$, and we called it as fourth type
Y-junction graph. It has $6l^{2}+30l+6+24ml$ vertices and $9l^{2}+33l+9+36ml$
edges. The carbon nanotube Y-junction graphs $J$, $J_{1}$, $J_{2}$, and
$J_{3}$ are shown in Figure $1$.

The edge partition of Y-junction graphs $J$, $J_{1}$, $J_{2}$, and $J_{3}$
based on the neighborhood degree-sum of end vertices of an edge is given in
Table $2$.

(a) Y-junction graph $J$

(b) Y-junction graph $J_{1}$

(c) Y-junction graph $J_{2}$

(d) Y-junction graph $J_{3}$

Figure 1: A symmetrical uncapped single-walled armchair carbon nanotubes Y-junction graphs Table 2: Edge partitions of $J$, $J_{1}$, $J_{2}$, and $J_{3}$ $d_{n}(u),d_{n}(v)$ |  $J$-frequency |  $J_{1}$-frequency |  $J_{2}$-frequency |  $J_{3}$-frequency
---|---|---|---|---
(3,7) | 0 | 4l | 8l | 12l
(5,5) | 6l | 4l | 2l | 0
(5,8) | 12l | 8l | 4l | 0
(7,7) | 0 | 2l | 4l | 6l
(7,9) | 0 | 4l | 8l | 12l
(8,8) | 6l | 4l | 2l | 0
(8,9) | 12l | 8l | 4l | 0
(9,9) | $9l^{2}-15l+36ml+9$ | $9l^{2}-9l+36ml+9$ | $9l^{2}-3l+36ml+9$ | $9l^{2}+3l+36ml+9$

##  5 NM-Polynomials and Topological Indices of Y-Junction Graphs

In this section, we develop the general expression of NM-polynomials for the
Y-junction graphs and then recover various neighborhood degree-sum based
topological indices from these polynomials.

######  Theorem 1.

Let $J$ be the Y-junction graph of an uncapped symmetrical single-walled
armchair carbon nanotube. Then
$NM(J;x,y)=6lx^{5}y^{5}+12lx^{5}y^{8}+6lx^{8}y^{8}+12lx^{8}y^{9}+(9l^{2}-15l+9+36ml)x^{9}y^{9}$.

###### Proof.

The Y-junction graph of an uncapped symmetrical single-walled armchair carbon
nanotubes has $9l^{2}+21l+9+36ml$ number of edges. Let $E_{(i,j)}$ be the set
of all edges with neighborhood degree sum of end vertices $i,j$, i.e.,
$E_{(i,j)}=\\{uv\in E(J):d_{n}(u)=i,\;d_{n}(v)=j\\}$.
By means of structural analysis of $J$, the edge set of $J$ can be partitioned
into five sets on the basis of neighborhood degree sum of end vertices as
follows:
$E_{(5,5)}=\\{uv\in E(J):d_{n}(u)=5,\;d_{n}(v)=5\\}$, $E_{(5,8)}=\\{uv\in
E(J):d_{n}(u)=5,\;d_{n}(v)=8\\}$, $E_{(8,8)}=\\{uv\in
E(J^{m}(n,n)):d_{n}(u)=8,\;d_{n}(v)=8\\}$, $E_{(8,9)}=\\{uv\in
E(J):d_{n}(u)=8,\;d_{n}(v)=9\\}$, $E_{(9,9)}=\\{uv\in
E(J):d_{n}(u)=9,\;d_{n}(v)=9\\}$, and $\lvert E_{(5,5)}\rvert=6l$, $\lvert
E_{(5,8)}\rvert=12l$, $\lvert E_{(8,8)}\rvert=6l$, $\lvert
E_{(8,9)}\rvert=12l$, $\lvert E_{(9,9)}\rvert=9l^{2}-15l+9+36ml$.

From Equation (1), the NM-polynomial of $J$ is obtained as follows:

$\displaystyle NM(J;x,y)$ $\displaystyle=$ $\displaystyle\sum\limits_{i\leq
j}\lvert E_{(i,j)}\rvert x^{i}y^{j}$ $\displaystyle=$ $\displaystyle\lvert
E_{(5,5)}\rvert x^{5}y^{5}+\lvert E_{(5,8)}\rvert x^{5}y^{8}+\lvert
E_{(8,8)}\rvert x^{8}y^{8}+\lvert E_{(8,9)}\rvert x^{8}y^{9}+\lvert
E_{(9,9)}\rvert x^{9}y^{9}$ $\displaystyle=$ $\displaystyle
6lx^{5}y^{5}+12lx^{5}y^{8}+6lx^{8}y^{8}+12lx^{8}y^{9}+(9l^{2}-15l+9+36ml)x^{9}y^{9}.$

∎

######  Theorem 2.

Let $J$ be the Y-junction graph of an uncapped symmetrical single-walled
armchair carbon nanotube . Then

(i) $NM_{1}(J)=162l^{2}+246l+648ml+162$

(ii) $NM_{2}(J)=729l^{2}+663l+2916ml+729$

(iii) $NF(J)=1458l^{2}+1446l+5832ml+1458$

(iv) ${}^{nm}M_{2}(J)=0.11l^{2}+0.62l+0.44ml+0.11$

(v)
$NR_{\alpha}(J)=6l(25^{\alpha}+2(40)^{\alpha}+64^{\alpha}+2(72)^{\alpha})+81^{\alpha}(9l^{2}-15l+9+36ml)$

(vi) $ND_{3}(J)=13122l^{2}+6702l+52488ml+13122$

(vii) $ND_{5}(J)=18l^{2}+44.86l+72ml+18$

(viii) $NH(J)=l^{2}+9.69l+4ml+1$

(ix) $NI(J)=40.5l^{2}+59.24l+162ml+40.5$

(x) $S(J)=1167.7l^{2}+714.23l+4670.9ml+1167.7$.

###### Proof.

Let
$f(x,y)=NM(J;x,y)=6lx^{5}y^{5}+12lx^{5}y^{8}+6lx^{8}y^{8}+12lx^{8}y^{9}+(9l^{2}-15l+9+36ml)x^{9}y^{9}$.
Then, we have
$D_{x}(f(x,y))=30lx^{5}y^{5}+60lx^{5}y^{8}+48lx^{8}y^{8}+96lx^{8}y^{9}+9(9l^{2}-15l+9+36ml)x^{9}y^{9}$.

$D_{y}(f(x,y))=30lx^{5}y^{5}+96lx^{5}y^{8}+48lx^{8}y^{8}+108lx^{8}y^{9}+9(9l^{2}-15l+9+36ml)x^{9}y^{9}$.

$D^{2}_{x}(f(x,y))=150lx^{5}y^{5}+300lx^{5}y^{8}+384lx^{8}y^{8}+768lx^{8}y^{9}+81(9l^{2}-15l+9+36ml)x^{9}y^{9}$.

$D^{2}_{y}(f(x,y))=150lx^{5}y^{5}+768lx^{5}y^{8}+384lx^{8}y^{8}+972lx^{8}y^{9}+81(9l^{2}-15l+9+36ml)x^{9}y^{9}$.

$D_{x}D_{y}(f(x,y))=150lx^{5}y^{5}+480lx^{5}y^{8}+384lx^{8}y^{8}+864lx^{8}y^{9}+81(9l^{2}-15l+9+36ml)x^{9}y^{9}$.

$(D_{x}+D_{y})f(x,y)=60lx^{5}y^{5}+156lx^{5}y^{8}+96lx^{8}y^{8}+204lx^{8}y^{9}+18(9l^{2}-15l+9+36ml)x^{9}y^{9}$.

$\displaystyle D_{x}D_{y}(D_{x}+D_{y})f(x,y)$ $\displaystyle=$ $\displaystyle
1500lx^{5}y^{5}+6240lx^{5}y^{8}+6144lx^{8}y^{8}+14688lx^{8}y^{9}+1458(9l^{2}-15l+$
$\displaystyle 9+36ml)x^{9}y^{9}.$

$(D^{2}_{x}+D^{2}_{y})f(x,y)=300lx^{5}y^{5}+1068lx^{5}y^{8}+768lx^{8}y^{8}+1740lx^{8}y^{9}+162(9l^{2}-15l+9+36ml)x^{9}y^{9}$.

$\displaystyle D_{x}^{\alpha}D_{y}^{\alpha}(f(x,y))$ $\displaystyle=$
$\displaystyle
6l(25)^{\alpha}x^{5}y^{5}+12l(40)^{\alpha}x^{5}y^{8}+6l(64)^{\alpha}x^{8}y^{8}+12l(72)^{\alpha}x^{8}y^{9}+(81)^{\alpha}$
$\displaystyle(9l^{2}-15l+9+36ml)x^{9}y^{9}.$

$S_{x}S_{y}(f(x,y))=\frac{6l}{25}x^{5}y^{5}+\frac{12l}{40}x^{5}y^{8}+\frac{6l}{64}x^{8}y^{8}+\frac{12l}{72}x^{8}y^{9}+\frac{(9l^{2}-15l+9+36ml)}{81}x^{9}y^{9}$.

$S_{y}D_{x}+S_{x}D_{y}(f(x,y))=12lx^{5}y^{5}+\frac{267l}{10}x^{5}y^{8}+12lx^{8}y^{8}+\frac{145l}{6}x^{8}y^{9}+2(9l^{2}-15l+9+36ml)x^{9}y^{9}$.

$2S_{x}T(f(x,y))=\frac{6l}{5}x^{10}+\frac{24l}{13}x^{13}+\frac{3l}{4}x^{16}+\frac{24l}{17}x^{17}+\frac{(9l^{2}-15l+9+36ml)}{9}x^{18}$.

$S_{x}TD_{x}D_{y}(f(x,y))=15lx^{10}+\frac{480l}{13}x^{13}+\frac{384l}{16}x^{16}+\frac{864l}{17}x^{17}+\frac{81(9l^{2}-15l+9+36ml)}{18}x^{18}$.

$S_{x}^{3}Q_{-2}TD_{x}^{3}D_{y}^{3}(f(x,y))=\frac{93750l}{512}x^{8}+\frac{768000l}{1331}x^{11}+\frac{1572864l}{2744}x^{14}+\frac{4478976l}{3375}x^{15}+\frac{531441(9l^{2}-15l+9+36ml)}{4096}x^{16}$.

Now, using Table $1$ we have

(i) $NM_{1}(J)=(D_{x}+D_{y})f(x,y)\rvert_{x=y=1}=162l^{2}+246l+648ml+162$.

(ii) $NM_{2}(J)=(D_{x}D_{y})f(x,y)\rvert_{x=y=1}=729l^{2}+663l+2916ml+729$.

(iii)
$NF(J)=(D^{2}_{x}+D^{2}_{y})f(x,y)\rvert_{x=y=1}=1458l^{2}+1446l+5832ml+1458$.

(iv)
${}^{nm}M_{2}(J)=(S_{x}S_{y})f(x,y)\rvert_{x=y=1}=0.11l^{2}+0.62l+0.44ml+0.11$.

(v)
$NR_{\alpha}(J)=(D^{\alpha}_{x}D^{\alpha}_{y})f(x,y)\rvert_{x=y=1}=6l(25^{\alpha}+2(40)^{\alpha}+64^{\alpha}+2(72)^{\alpha})+81^{\alpha}(9l^{2}-15l+9+36ml)$.

(vi)
$ND_{3}(J)=D_{x}D_{y}(D_{x}+D_{y})f(x,y)\rvert_{x=y=1}=13122l^{2}+6702l+52488ml+13122$.

(vii)
$ND_{5}(J)=S_{y}D_{x}+S_{x}D_{y}(f(x,y))\rvert_{x=y=1}=18l^{2}+44.86l+72ml+18$.

(viii) $NH(J)=2S_{x}T(f(x,y))\rvert_{x=y=1}=l^{2}+9.69l+4ml+1$.

(ix)
$NI(J)=S_{x}TD_{x}D_{y}(f(x,y))\rvert_{x=y=1}=40.5l^{2}+59.24l+162ml+40.5$.

(x)
$S(J)=S_{x}^{3}Q_{-2}TD_{x}^{3}D_{y}^{3}(f(x,y))\rvert_{x=y=1}=1167.7l^{2}+714.23l+4670.9ml+1167.7$.
∎

######  Theorem 3.

Let $J_{1}$ be the second type Y-junction graph of an uncapped symmetrical
single-walled armchair carbon nanotube. Then
$NM(J_{1};x,y)=4lx^{3}y^{7}+4lx^{5}y^{5}+8lx^{5}y^{8}+2lx^{7}y^{7}+4lx^{7}y^{9}+4lx^{8}y^{8}+8lx^{8}y^{9}+(9l^{2}-9l+9+36ml)x^{9}y^{9}$.

###### Proof.

The second type Y-junction graph of an uncapped symmetrical single-walled
armchair carbon nanotubes has $9l^{2}+25l+9+36ml$ edges. Let $E_{(i,j)}$ be
the set of all edges with neighborhood degree sum of end vertices $i,j$, i.e.,
$E_{(i,j)}=\\{uv\in E(J_{1}):d_{n}(u)=i,\;d_{n}(v)=j\\}$.
By means of structure analysis of $J_{1}$, the edge set of $J_{1}$ can be
partitioned into eight sets on the basis of neighborhood degree sum of end
vertices as follows:
$E_{(3,7)}=\\{uv\in E(J_{1}):d_{n}(u)=3,\;d_{n}(v)=7\\}$, $E_{(5,5)}=\\{uv\in
E(J_{1}):d_{n}(u)=5,\;d_{n}(v)=5\\}$, $E_{(5,8)}=\\{uv\in
E(J_{1}):d_{n}(u)=5,\;d_{n}(v)=8\\}$, $E_{(7,7)}=\\{uv\in
E(J_{1}):d_{n}(u)=7,\;d_{n}(v)=7\\}$, $E_{(7,9)}=\\{uv\in
E(J_{1}):d_{n}(u)=7,\;d_{n}(v)=9\\}$, $E_{(8,8)}=\\{uv\in
E(J_{1}):d_{n}(u)=8,\;d_{n}(v)=8\\}$, $E_{(8,9)}=\\{uv\in
E(J_{1}):d_{n}(u)=8,\;d_{n}(v)=9\\}$, $E_{(9,9)}=\\{uv\in
E(J_{1}):d_{n}(u)=9,\;d_{n}(v)=9\\}$, and $\lvert E_{(3,7)}\rvert=4l$, $\lvert
E_{(5,5)}\rvert=4l$, $\lvert E_{(5,8)}\rvert=8l$, $\lvert E_{(7,7)}\rvert=2l$,
$\lvert E_{(7,9)}\rvert=4l$, $\lvert E_{(8,8)}\rvert=4l$, $\lvert
E_{(8,9)}\rvert=8l$, $\lvert E_{(9,9)}\rvert=9l^{2}-9l+9+36ml$.
From Equation (1), the NM-polynomial of $J_{1}$ is obtained as follows:

$\displaystyle NM(J_{1};x,y)$ $\displaystyle=$
$\displaystyle\sum\limits_{i\leq j}\lvert E_{(i,j)}\rvert x^{i}y^{j}$
$\displaystyle=$ $\displaystyle\lvert E_{(3,7)}\rvert x^{3}y^{7}+\lvert
E_{(5,5)}\rvert x^{5}y^{5}+\lvert E_{(5,8)}\rvert x^{5}y^{8}+\lvert
E_{(7,7)}\rvert x^{7}y^{7}+\lvert E_{(7,9)}\rvert x^{7}y^{9}+$
$\displaystyle\lvert E_{(8,8)}\rvert x^{8}y^{8}+\lvert E_{(8,9)}\rvert
x^{8}y^{9}+\lvert E_{(9,9)}\rvert x^{9}y^{9}$ $\displaystyle=$ $\displaystyle
4lx^{3}y^{7}+4lx^{5}y^{5}+8lx^{5}y^{8}+2lx^{7}y^{7}+4lx^{7}y^{9}+4lx^{8}y^{8}+8lx^{8}y^{9}+$
$\displaystyle(9l^{2}-9l+9+36ml)x^{9}y^{9}.$

∎

######  Theorem 4.

Let $J_{1}$ be the second type Y-junction graph of an uncapped symmetrical
single-walled armchair carbon nanotube. Then
(i) $NM_{1}(J_{1})=162l^{2}+314l+648ml+162$

(ii) $NM_{2}(J_{1})=729l^{2}+957l+2916ml+729$

(iii) $NF(J_{1})=1458l^{2}+2074l+5832ml+1458$

(iv) ${}^{nm}M_{2}(J_{1})=0.11l^{2}+0.72l+0.44ml+0.11$

$\displaystyle(v)\;NR_{\alpha}(J_{1})$ $\displaystyle=$ $\displaystyle
2l(2(21)^{\alpha}+2(25)^{\alpha}+4(40)^{\alpha}+(49)^{\alpha}+2(63)^{\alpha}+2(64)^{\alpha}+4(72)^{\alpha})+(81)^{\alpha}(9l^{2}-9l+9$
$\displaystyle+36ml)$

(vi) $ND_{3}(J_{1})=13122l^{2}+12170l+52488ml+13122$

(vii) $ND_{5}(J_{1})=18l^{2}+56.328l+72ml+18$

(viii) $NH(J_{1})=l^{2}+3.98l+4ml+1$

(ix) $NI(J_{1})=40.5l^{2}+75.15l+162ml+40.5$

(x) $S(J_{1})=1167.7l^{2}+1178.92l+4670.9ml+1167.7$.

###### Proof.

Refer to Theorem $2$ for proof. ∎

######  Theorem 5.

Let $J_{2}$ be the third type Y-junction graph of an uncapped symmetrical
single-walled armchair carbon nanotube. Then
$NM(J_{2};x,y)=8lx^{3}y^{7}+2lx^{5}y^{5}+4lx^{5}y^{8}+4lx^{7}y^{7}+8lx^{7}y^{9}+2lx^{8}y^{8}+4lx^{8}y^{9}+(9l^{2}-3l+9+36ml)x^{9}y^{9}$.

###### Proof.

The third type Y-junction graph of an uncapped symmetrical single-walled
armchair carbon nanotubes has $9l^{2}+29l+9+36ml$ number of edges. Let
$E_{(i,j)}$ be the set of all edges with neighborhood degree sum of end
vertices $i,j$, i.e., $E_{(i,j)}=\\{uv\in
E(J_{2}):d_{n}(u)=i,\;d_{n}(v)=j\\}$.
By means of structure analysis of $J_{2}$, the edge set of $J_{2}$ can be
partitioned into eight sets on the basis of neighborhood degree sum of end
vertices as follows:
$E_{(3,7)}=\\{uv\in E(J_{2}):d_{n}(u)=3,\;d_{n}(v)=7\\}$, $E_{(5,5)}=\\{uv\in
E(J_{2}):d_{n}(u)=5,\;d_{n}(v)=5\\}$, $E_{(5,8)}=\\{uv\in
E(J_{2}^{)}:d_{n}(u)=5,\;d_{n}(v)=8\\}$, $E_{(7,7)}=\\{uv\in
E(J_{2}):d_{n}(u)=7,\;d_{n}(v)=7\\}$, $E_{(7,9)}=\\{uv\in
E(J_{2}):d_{n}(u)=7,\;d_{n}(v)=9\\}$, $E_{(8,8)}=\\{uv\in
E(J_{2}):d_{n}(u)=8,\;d_{n}(v)=8\\}$, $E_{(8,9)}=\\{uv\in
E(J_{2}):d_{n}(u)=8,\;d_{n}(v)=9\\}$, $E_{(9,9)}=\\{uv\in
E(J_{2}):d_{n}(u)=9,\;d_{n}(v)=9\\}$, and $\lvert E_{(3,7)}\rvert=8l$, $\lvert
E_{(5,5)}\rvert=2l$, $\lvert E_{(5,8)}\rvert=4l$, $\lvert E_{(7,7)}\rvert=4l$,
$\lvert E_{(7,9)}\rvert=8l$, $\lvert E_{(8,8)}\rvert=2l$, $\lvert
E_{(8,9)}\rvert=4l$, $\lvert E_{(9,9)}\rvert=9l^{2}-3l+9+36ml$.

From Equation (1), the NM-polynomial of $J_{2}$ is obtained as follows:

$\displaystyle NM(J_{2};x,y)$ $\displaystyle=$
$\displaystyle\sum\limits_{i\leq j}\lvert E_{(i,j)}\rvert x^{i}y^{j}$
$\displaystyle=$ $\displaystyle\lvert E_{(3,7)}\rvert x^{3}y^{7}+\lvert
E_{(5,5)}\rvert x^{5}y^{5}+\lvert E_{(5,8)}\rvert x^{5}y^{8}+\lvert
E_{(7,7)}\rvert x^{7}y^{7}+\lvert E_{(7,9)}\rvert x^{7}y^{9}+$
$\displaystyle\lvert E_{(8,8)}\rvert x^{8}y^{8}+\lvert E_{(8,9)}\rvert
x^{8}y^{9}+\lvert E_{(9,9)}\rvert x^{9}y^{9}$ $\displaystyle=$ $\displaystyle
8lx^{3}y^{7}+2lx^{5}y^{5}+4lx^{5}y^{8}+4lx^{7}y^{7}+8lx^{7}y^{9}+2lx^{8}y^{8}+4lx^{8}y^{9}+$
$\displaystyle(9l^{2}-3l+9+36ml)x^{9}y^{9}.$

∎

######  Theorem 6.

Let $J_{2}$ be the third type Y-junction graph of an uncapped symmetrical
single-walled armchair carbon nanotube. Then

(i) $NM_{1}(J_{2})=162l^{2}+382l+648ml+162$

(ii) $NM_{2}(J_{2})=729l^{2}+1251l+2916ml+729$

(iii) $NF(J_{2})=1458l^{2}+2478l+5832ml+1458$

(iv) ${}^{nm}M_{2}(J_{2})=0.11l^{2}+0.819l+0.44ml+0.11$

(v)
$NR_{\alpha}(J_{2})=2l(4(21)^{\alpha}+(25)^{\alpha}+2(40)^{\alpha}+2(49)^{\alpha}+4(63)^{\alpha}+(64)^{\alpha}+2(72)^{\alpha})+(81)^{\alpha}(9l^{2}-3l+9+36ml)$

(vi) $ND_{3}(J_{2})=13122l^{2}+17638l+52488ml+13122$

(vii) $ND_{5}(J_{2})=18l^{2}+65.56l+72ml+18$

(viii) $NH(J_{2})=l^{2}+4.57l+4ml+1$

(ix) $NI(J_{2})=40.5l^{2}+91.048l+162ml+40.5$

(x) $S(J_{2})=1167.7l^{2}+1643.61l+4670.9ml+1167.7$.

###### Proof.

Refer to Theorem $2$ for proof. ∎

######  Theorem 7.

Let $J_{3}$ be the fourth type Y-junction graph of an uncapped symmetrical
single-walled armchair carbon nanotube. Then
$NM(J_{3};x,y)=12lx^{3}y^{7}+6lx^{7}y^{7}+12lx^{7}y^{9}+(9l^{2}+3l+9+36ml)x^{9}y^{9}$.

###### Proof.

The fourth type Y-junction graph of an uncapped symmetrical single-walled
armchair carbon nanotube has $9l^{2}+33l+9+36ml$ number of edges. Let
$E_{(i,j)}$ be the set of all edges with neighborhood degree sum of end
vertices $i,j$, i.e., $E_{(i,j)}=\\{uv\in
E(J_{3}):d_{n}(u)=i,\;d_{n}(v)=j\\}$.
By means of structure analysis of $J_{3}$, the edge set of $J_{3}$ can be
partitioned into four sets on the basis of neighborhood degree sum of end
vertices as follows:
$E_{(3,7)}=\\{uv\in E(J_{3}):d_{n}(u)=3,\;d_{n}(v)=7\\}$, $E_{(7,7)}=\\{uv\in
E(J_{3}):d_{n}(u)=7,\;d_{n}(v)=7\\}$, $E_{(7,9)}=\\{uv\in
E(J_{3}):d_{n}(u)=7,\;d_{n}(v)=9\\}$, $E_{(9,9)}=\\{uv\in
E(J_{3}):d_{n}(u)=9,\;d_{n}(v)=9\\}$, and $\lvert E_{(3,7)}\rvert=12l$,
$\lvert E_{(7,7)}\rvert=6l$, $\lvert E_{(7,9)}\rvert=12l$, $\lvert
E_{(9,9)}\rvert=9l^{2}+3l+9+36ml$.

From Equation (1), the NM-polynomial of $J_{3}$ is obtained as follows:

$\displaystyle NM(J_{3};x,y)$ $\displaystyle=$
$\displaystyle\sum\limits_{i\leq j}\lvert E_{(i,j)}\rvert x^{i}y^{j}$
$\displaystyle=$ $\displaystyle\lvert E_{(3,7)}\rvert x^{3}y^{7}+\lvert
E_{(7,7)}\rvert x^{7}y^{7}+\lvert E_{(7,9)}\rvert x^{7}y^{9}+\lvert
E_{(9,9)}\rvert x^{9}y^{9}$ $\displaystyle=$ $\displaystyle
12lx^{3}y^{7}+6lx^{7}y^{7}+12lx^{7}y^{9}+(9l^{2}+3l+9+36ml)x^{9}y^{9}.$

∎

######  Theorem 8.

Let $J_{3}$ be the fourth type Y-junction graph of an uncapped symmetrical
single-walled armchair carbon nanotube. Then

(i) $NM_{1}(J_{3})=162l^{2}+450l+648ml+162$

(ii) $NM_{2}(J_{3})=729l^{2}+1545l+2916ml+729$

(iii) $NF(J_{3})=1458l^{2}+3330l+5832ml+1458$

(iv) ${}^{nm}M_{2}(J_{3})=0.11l^{2}+0.92l+0.44ml+0.11$

(v)
$NR_{\alpha}(J_{3})=6l(2(21)^{\alpha}+(49)^{\alpha}+2(63)^{\alpha})+(81)^{\alpha}(9l^{2}+3l+9+36ml)$

(vi) $ND_{3}(J_{3})=13122l^{2}+23106l+52488ml+13122$

(vii) $ND_{5}(J_{3})=18l^{2}+75.90l+72ml+18$

(viii) $NH(J_{3})=l^{2}+5.090l+4ml+1$

(ix) $NI(J_{3})=40.5l^{2}+106.95l+162ml+40.5$

(x) $S(J_{3})=1167.7l^{2}+2085.95l+4670.9ml+1167.7$.

###### Proof.

Refer to Theorem $2$ for proof. ∎

##  6 Graph Index-Entropies of Y-Junction Graphs

In this section, we compute the index-entropy of carbon nanotube Y-junctions
in terms of neighborhood degree sum-based topological indices. We first
compute index-entropies of the Y-junction graph $J$ whose edge partition is
given in Table $2$.

  * •

Third-version of Zagreb index-entropy of $J$

From part (i) of Theorem $2$, we have

$NM_{1}(J)=162l^{2}+246l+648ml+162.$ (19)

Now, from Equation ($4$), the third-version of Zagreb index-entropy of $J$ is

$H_{\beta_{1}}(J)=log(NM_{1}(J))-\frac{1}{NM_{1}(J)}\sum\limits_{e\in
E(J)}{\beta}_{1}(e)log\beta_{1}(e).$ (20)

Using Table $2$ and Equation ($19$) in Equation ($20$), we get the required
third-version of Zagreb index-entropy of $J$ as follows:

$\displaystyle H_{\beta_{1}}(J)$ $\displaystyle=$ $\displaystyle
log(NM_{1}(J))-\frac{1}{NM_{1}(J)}\sum\limits_{e\in
E(J)}{\beta}_{1}(e)log\beta_{1}(e)$ $\displaystyle=$ $\displaystyle
log(162l^{2}+246l+648ml+162)-\frac{1}{162l^{2}+246l+648ml+162}\bigg{(}6l(10)(log10)+$
$\displaystyle
12l(13)(log13)+6l(16)(log16)+12l(17)(log17)+(9l^{2}-15l+36ml+9)(18)(log18)\bigg{)}$
$\displaystyle=$ $\displaystyle
log(162l^{2}+246l+648ml+162)-\frac{1}{162l^{2}+246l+648ml+162}\bigg{(}60l(log10)+$
$\displaystyle
156l(log13)+96l(log16)+204l(log17)+(162l^{2}-270l+648ml+162)(log18)\bigg{)}$
$\displaystyle=$ $\displaystyle
log(162l^{2}+246l+648ml+162)-\frac{1}{162l^{2}+246l+648ml+162}\bigg{(}60l(1)+156l(1.1139433523)$
$\displaystyle+96l(1.2041199827)+204l(1.2304489214)+(162l^{2}-270l+648ml+162)(1.2552725051)\bigg{)}$
$\displaystyle\approx
log(162l^{2}+246l+648ml+162)-\frac{202.5l^{2}+261.78l+810ml+202.5}{162l^{2}+246l+648ml+162}.$

  * •

Neighborhood second Zagreb index-entropy of $J$

From part (ii) of Theorem $2$, we have

$NM_{2}(J)=729l^{2}+663l+2916ml+729.$ (21)

By using the values given in Table $2$ and Equation ($21$) in Equation ($6$),
we get the required neighborhood second Zagreb index-entropy of $J$ as
follows:

$\displaystyle H_{\beta_{2}}(J)$ $\displaystyle=$ $\displaystyle
log(NM_{2}(J))-\frac{1}{NM_{2}(J)}\sum\limits_{e\in
E(J)}{\beta}_{2}(e)log\beta_{2}(e)$ $\displaystyle=$ $\displaystyle
log(729l^{2}+663l+2916ml+729)-\frac{1}{729l^{2}+663l+2916ml+729}\bigg{(}6l(25)(log25)+$
$\displaystyle
12l(40)(log40)+6l(64)(log64)+12l(72)(log72)+(9l^{2}-15l+36ml+9)(81)(log81)\bigg{)}$
$\displaystyle\approx
log(729l^{2}+663l+2916ml+729)-\frac{1391.22l^{2}+958.27l+5564.88ml+1391.22}{729l^{2}+663l+2916ml+729}.$

Similarly, we compute the remaning index-entropies of $J$. Table $3$ shows
some calculated graph index-entropies of $J$.

In this way, the topological index-based entropies for Y-junction graphs
$J_{1}$, $J_{2}$, and $J_{3}$ are calculated.
The index-based entropies of $J_{1}$ $J_{2}$, and $J_{3}$ are given in Tables
$4$, $5$, and $6$.

Table 3: Index-entropies of $J$ Entropy | Values of entropies
---|---
$H_{\beta_{3}}(J)$ | log$(1458l^{2}+1446l+5832ml+1458)-\frac{3221.62l^{2}+2601.62l+12885.84ml+3221.46}{1458l^{2}+1446l+5832ml+1458}$
$H_{\beta_{4}}(J)$ | log$(0.11l^{2}+0.62l+0.44ml+0.11)+\frac{0.207l^{2}+0.92l+0.828ml+0.207}{0.11l^{2}+0.62l+0.44ml+0.11}$
$H_{\beta_{5}}(J)$ | log$(13122l^{2}+12170l+52488ml+13122)-\frac{41514.75l^{2}+15202.13l+166059.31ml+41514.75}{13122l^{2}+12170l+52488ml+13122}$
$H_{\beta_{6}}(J)$ | log$(18l^{2}+44.86l+72ml+18)-\frac{5.41l^{2}+14.81l+21.67ml+5.41}{18l^{2}+44.86l+72ml+18}$
$H_{\beta_{7}}(J)$ | log$(l^{2}+9.69l+4ml+1)+\frac{0.95l^{2}+2.72l+3.81ml+0.95}{l^{2}+9.69l+4ml+1}$
$H_{\beta_{8}}(J)$ | log$(40.5l^{2}+59.24l+162ml+40.5)-\frac{26.45l^{2}+26.18l+105.82ml+26.45}{40.5l^{2}+59.24l+162ml+40.5}$
Table 4: Index-entropies of $J_{1}$ Entropy | Values of entropies
---|---
$H_{\beta_{1}}(J_{1})$ | log$(162l^{2}+314l+648ml+162)-\frac{203.31l^{2}+346.09l+813.24ml+203.31}{162l^{2}+314l+648ml+162}$
$H_{\beta_{2}}(J_{1})$ | log$(729l^{2}+957l+2916ml+729)-\frac{1391.22l^{2}+1523.54l+5564.88ml+1391.22}{729l^{2}+957l+2916ml+729}$
$H_{\beta_{3}}(J_{1})$ | log$(1458l^{2}+2074l+5832ml+1458)-\frac{3221.46l^{2}+3991l+12885.84ml+3221.46}{1458l^{2}+2074l+5832ml+1458}$
$H_{\beta_{4}}(J_{1})$ | log$(0.11l^{2}+0.72l+0.44ml+0.11)+\frac{0.207l^{2}+1.065l+0.897ml+0.207}{0.11l^{2}+0.72l+0.44ml+0.11}$
$H_{\beta_{5}}(J_{1})$ | log$(13122l^{2}+12170l+52488ml+13122)-\frac{41514.75l^{2}+32699.53l+166059ml+41514.75}{13122l^{2}+12170l+52488ml+13122}$
$H_{\beta_{6}}(J_{1})$ | log$(18l^{2}+56.328l+72ml+18)-\frac{5.41l^{2}+19.09l+21.67ml+5.41}{18l^{2}+56.328l+72ml+18}$
$H_{\beta_{7}}(J_{1})$ | log$(l^{2}+3.98l+4ml+1)+\frac{0.9l^{2}+3.21l+3.6ml+0.9}{l^{2}+3.98l+4ml+1}$
$H_{\beta_{8}}(J_{1})$ | log$(40.5l^{2}+75.15l+162ml+40.5)-\frac{26.37l^{2}+36.84l+105.48ml+26.37}{40.5l^{2}+75.15l+162ml+40.5}$
Table 5: Index-entropies of $J_{2}$ Entropy | Values of entropies
---|---
$H_{\beta_{1}}(J_{2})$ | log$(162l^{2}+382l+648ml+162)-\frac{203.31l^{2}+430.65l+813.24ml+203.31}{162l^{2}+382l+648ml+162}$
$H_{\beta_{2}}(J_{2})$ | log$(729l^{2}+1251l+2916ml+729)-\frac{1391.22l^{2}+2088.77l+5564.88ml+1391.22}{729l^{2}+1251l+2916ml+729}$
$H_{\beta_{3}}(J_{2})$ | log$(1458l^{2}+2478l+5832ml+1458)-\frac{3221.46l^{2}+5380.37l+12885.84ml+3221.46}{1458l^{2}+2478l+5832ml+1458}$
$H_{\beta_{4}}(J_{2})$ | log$(0.11l^{2}+0.819l+0.44ml+0.11)+\frac{0.099l^{2}+1.007l+0.396ml+0.099}{0.11l^{2}+0.819l+0.44ml+0.11}$
$H_{\beta_{5}}(J_{2})$ | log$(13122l^{2}+17638l+52488ml+13122)-\frac{41514.75l^{2}+50196.95l+166059ml+50196.95}{13122l^{2}+17638l+52488ml+13122}$
$H_{\beta_{6}}(J_{2})$ | log$(18l^{2}+65.56l+72ml+18)-\frac{5.41l^{2}+23.44l+21.67ml+5.41}{18l^{2}+65.56l+72ml+18}$
$H_{\beta_{7}}(J_{2})$ | log$(l^{2}+4.57l+4ml+1)+\frac{0.9l^{2}+3.61l+3.6ml+0.9}{l^{2}+4.57l+4ml+1}$
$H_{\beta_{8}}(J_{2})$ | log$(40.5l^{2}+91.048l+162ml+40.5)-\frac{26.37l^{2}+46.387l+105.48ml+26.37}{40.5l^{2}+91.048l+162ml+40.5}$
Table 6: Index-entropies of $J_{3}$ Entropy | Values of entropies
---|---
$H_{\beta_{1}}(J_{3})$ | log$(162l^{2}+450l+648ml+162)-\frac{203.31l^{2}+515.23l+813.24ml+203.31}{162l^{2}+450l+648ml+162}$
$H_{\beta_{2}}(J_{3})$ | log$(729l^{2}+1545l+2916ml+729)-\frac{1391.22l^{2}+2654.14l+5564.88ml+1391.22}{729l^{2}+1545l+2916ml+729}$
$H_{\beta_{3}}(J_{3})$ | log$(1458l^{2}+3330l+5832ml+1458)-\frac{3221.46l^{2}+6769.75l+12885.84ml+3221.46}{1458l^{2}+3330l+5832ml+1458}$
$H_{\beta_{4}}(J_{3})$ | log$(0.11l^{2}+0.92l+0.44ml+0.11)+\frac{0.18l^{2}+1.35l+0.72ml+0.18}{0.11l^{2}+0.92l+0.44ml+0.11}$
$H_{\beta_{5}}(J_{3})$ | log$(13122l^{2}+23106l+52488ml+13122)-\frac{41514.75l^{2}+67694.4l+166059ml+41514}{13122l^{2}+23106l+52488ml+13122}$
$H_{\beta_{6}}(J_{3})$ | log$(18l^{2}+75.90l+72ml+18)-\frac{5.41l^{2}+27.79l+21.67ml+5.41}{18l^{2}+75.90l+72ml+18}$
$H_{\beta_{7}}(J_{3})$ | log$(l^{2}+5.090l+4ml+1)+\frac{0.9l^{2}+4.04l+3.6ml+0.9}{l^{2}+5.090l+4ml+1}$
$H_{\beta_{8}}(J_{3})$ | log$(40.5l^{2}+106.95l+162ml+40.5)-\frac{26.37l^{2}+56.44l+105.48ml+26.37}{40.5l^{2}+106.95l+162ml+40.5}$

##  7 Numerical Results and Discussions

The numerical values of topological indices and graph index-entropies of
Y-junction graphs are computed in this section for some values of $l$ and $m$.
In addition, we plot line and bar graphs for comparison of the obtained
results. Here, we use the logarithm of the base $10$ for calculations.

The numerical values of topological indices for Y-junction graph $J$ are given
in Table $7$. The logarithmic values of Table $7$ are plotted in Figure $2$.
From the vertical axis of Figure $2$, we can conclude that for Y-junction
graph $J$, the topological indices have the following order: ${}^{nm}M_{2}\leq
NR_{-1/2}\leq NH\leq ND_{5}\leq NI\leq NM_{1}\leq NM_{2}\leq S\leq NF\leq
ND_{3}$. The third NDe index has the most dominating nature compared to other
topological indices, whereas neighborhood second modified Zagreb index grew
slowly.

Table 7: Numerical values of topological indices for Y-junction graph $J$ [$l,m$] | $NM_{1}(J)$ | $NM_{2}(J)$ | $NF(J)$ | $N^{m}M_{2}(J)$ | $NR_{-\frac{1}{2}}(J)$ | $ND_{3}(J)$ | $ND_{5}(J)$ | $NH(J)$ | $NI(J)$ | $S(J)$
---|---|---|---|---|---|---|---|---|---|---
[2,2] | 3894 | 16635 | 33510 | 3.55 | 20.7436 | 288966 | 467.72 | 40.38 | 968.92 | 25950.56
[3,3] | 8190 | 35523 | 71406 | 6.92 | 45.27693 | 623718 | 962.58 | 75.07 | 2040.63 | 55857.79
[4,4] | 14106 | 61701 | 123882 | 11.39 | 79.81026 | 1089690 | 1637.44 | 119.76 | 3517.34 | 97442.22
[5,5] | 21642 | 95169 | 190938 | 16.96 | 124.3436 | 1686882 | 2492.3 | 174.45 | 5399.05 | 150703.9
[6,6] | 30798 | 135927 | 272574 | 23.63 | 178.8769 | 2415294 | 3527.16 | 239.14 | 7685.76 | 215642.7
[7,7] | 41574 | 183975 | 368790 | 31.4 | 243.4103 | 3274926 | 4742.02 | 313.83 | 10377.47 | 292258.7
[8,8] | 53970 | 239313 | 479586 | 40.27 | 317.9436 | 4265778 | 6136.88 | 398.52 | 13474.18 | 380551.9
[9.9] | 67986 | 301941 | 604962 | 50.24 | 440.4769 | 5387850 | 7711.74 | 493.21 | 16975.89 | 480522.4
[10,10] | 83622 | 371859 | 744918 | 61.31 | 497.0103 | 6641142 | 9466.6 | 597.9 | 20882.6 | 592170
Figure 2: Graphical comparison among topological indices of Y-junction graph
$J$

Table $8$ shows some numerical values of topological indices for Y-junction
graph $J_{1}$. The logarithmic values of these topological indices are plotted
in Figure $3$. From Figure $3$, we can conclude that the topological indices
for Y-junction graph $J_{1}$ have the following order: ${}^{nm}M_{2}\leq
NH\leq NR_{-1/2}\leq ND_{5}\leq NI\leq NM_{1}\leq NM_{2}\leq S\leq NF\leq
ND_{3}$. Also, we see that the logarithemic values of $NR_{-1/2}$ and $NH$ for
$J_{1}$ are almost same.

Table 8: Numerical values of topological indices for Y-junction graph $J_{1}$ [$l,m$] | $NM_{1}(J_{1})$ | $NM_{2}(J_{1})$ | $NF(J_{1})$ | ${}^{nm}M_{2}(J_{1})$ | $NR_{-\frac{1}{2}}(J_{1})$ | $ND_{3}(J_{1})$ | $ND_{5}(J_{1})$ | $NH(J_{1})$ | $NI(J_{1})$ | $S(J_{1})$
---|---|---|---|---|---|---|---|---|---|---
[2,2] | 4030 | 17223 | 35166 | 3.75 | 29.34052 | 299902 | 490.656 | 28.96 | 1000.8 | 26879.94
[3,3] | 8394 | 36405 | 74190 | 7.22 | 58.51078 | 640122 | 996.984 | 57.94 | 2088.45 | 57251.86
[4,4] | 14378 | 62877 | 127994 | 11.79 | 97.68103 | 1111562 | 1683.312 | 96.92 | 3581.1 | 99300.98
[5,5] | 21982 | 96639 | 196578 | 17.46 | 146.8513 | 1714222 | 2549.64 | 145.9 | 5478.75 | 153027.3
[6,6] | 31206 | 137691 | 279942 | 24.23 | 206.0216 | 2448102 | 3595.968 | 204.88 | 7781.4 | 218430.8
[7,7] | 42050 | 186033 | 378086 | 32.1 | 275.1918 | 3313202 | 4822.296 | 273.86 | 10489.05 | 295511.5
[8,8] | 54514 | 241665 | 491010 | 41.07 | 354.3621 | 4309522 | 6228.624 | 352.84 | 13601.7 | 384269.5
[9.9] | 68598 | 304587 | 618714 | 51.14 | 443.5323 | 5437062 | 7814.952 | 441.82 | 17119.35 | 484704.6
[10,10] | 84302 | 374799 | 761198 | 62.31 | 542.7026 | 6695822 | 9581.28 | 540.8 | 21042 | 596816.9
Figure 3: Graphical comparison among topological indices of Y-junction graph
$J_{1}$

Table $9$ shows some calculated values of topological indices for Y-junction
graph $J_{2}$. The logarithmic values of these indices are plotted in Figure
$4$. The vertical axis of Figure $4$ shows the comparison clearly. Figure $4$
shows that the logarithmic values of $ND_{3}$ are extremely high when compared
to other topological indices of $J_{2}$. From Figure $4$, we see that the
graph of $NR_{-1/2}$ and $NH$ are almost coincide.

Table 9: Numerical values of topological indices for Y-junction graph $J_{2}$ [$l,m$] | $NM_{1}(J_{2})$ | $NM_{2}(J_{2})$ | $NF(J_{2})$ | ${}^{nm}M_{2}(J_{2})$ | $NR_{-\frac{1}{2}}(J_{2})$ | $ND_{3}(J_{2})$ | $ND_{5}(J_{2})$ | $NH(J_{2})$ | $NI(J_{2})$ | $S(J_{2})$
---|---|---|---|---|---|---|---|---|---|---
[2,2] | 4166 | 17811 | 35574 | 3.948 | 30.49121 | 310838 | 509.12 | 30.14 | 1032.596 | 27809.32
[3,3] | 8598 | 37287 | 74502 | 7.517 | 60.23681 | 656526 | 1024.68 | 59.71 | 2136.144 | 58645.93
[4,4] | 14650 | 64053 | 128010 | 12.186 | 99.98241 | 1133434 | 1720.24 | 99.28 | 3644.692 | 101159.7
[5,5] | 22322 | 98109 | 196098 | 17.955 | 149.728 | 1741562 | 2595.8 | 148.85 | 5558.24 | 155350.8
[6,6] | 31614 | 139455 | 278766 | 24.824 | 209.2192 | 2480910 | 3651.36 | 208.42 | 7876.788 | 221219
[7,7] | 42526 | 188091 | 376014 | 32.793 | 279.2192 | 3351478 | 4886.92 | 277.99 | 10600.34 | 298764.4
[8,8] | 55058 | 244017 | 487842 | 41.862 | 358.9648 | 4353266 | 6302.48 | 357.56 | 13728.88 | 387987
[9,9] | 69210 | 307233 | 614250 | 52.031 | 448.7104 | 5486274 | 7898.48 | 447.13 | 17262.43 | 488886.8
[10,10] | 84982 | 377739 | 755238 | 63.3 | 548.456 | 6750502 | 9673.6 | 546.7 | 21200.98 | 601463.8
Figure 4: Graphical comparison among topological indices of Y-junction $J_{2}$

Table $10$ shows some numerical values of topological indices of Y-junction
$J_{3}$. Figure $5$ depicts the graphical comparison of these indices. Table
$10$ and Figure $5$ show that the values of topological indices strictly
increase as the values of $l$ and $m$ increases.
From Tables $7$, $8$, $9$, and $10$, we see that as the values of $l$ and $m$
in Y-junction graphs increases, the corresponding values of topological
indices grew very fastly.

Table 10: Numerical values of topological indices of Y-junction graph $J_{3}$ [$l,m$] | $NM_{1}(J_{3})$ | $NM_{2}(J_{3})$ | $NF(J_{3})$ | $N^{m}M_{2}(J_{3})$ | $NR_{-\frac{1}{2}}(J_{3})$ | $ND_{3}(J_{3})$ | $ND_{5}(J_{3})$ | $NH(J_{3})$ | $NI(J_{3})$ | $S(J_{3})$
---|---|---|---|---|---|---|---|---|---|---
[2,2] | 4302 | 18399 | 37278 | 4.15 | 31.6419 | 321774 | 529.8 | 31.18 | 1064.4 | 28694
[3,3] | 8802 | 38169 | 77058 | 7.82 | 61.9628 | 672930 | 1055.7 | 61.27 | 2183.85 | 59973
[4,4] | 14922 | 65229 | 131418 | 12.59 | 102.284 | 1155306 | 1761.6 | 101.36 | 3708.3 | 102929
[5,5] | 22662 | 99579 | 200358 | 18.46 | 152.605 | 1768902 | 2647.5 | 151.45 | 5637.75 | 157562
[6,6] | 32022 | 141219 | 283878 | 25.43 | 212.926 | 2513718 | 3713.4 | 211.54 | 7972.2 | 223873
[7,7] | 43002 | 190149 | 381978 | 33.5 | 283.247 | 3389754 | 4959.3 | 281.63 | 10711.7 | 301861
[8,8] | 55602 | 246369 | 494658 | 42.67 | 363.568 | 4397010 | 6385.2 | 361.72 | 13856.1 | 391526
[9,9] | 69822 | 309879 | 621918 | 52.94 | 453.889 | 5535486 | 7991.1 | 451.81 | 17405.6 | 492868
[10,10] | 85662 | 380679 | 763758 | 64.31 | 554.209 | 6805182 | 9777 | 551.9 | 21360 | 605887
Figure 5: Graphical comparison among topological indices of Y-junction graph
$J_{3}$

A few values of graph index-entropies of Y-junction graph $J$ are listed in
Table $11$ and illustrated in Figure $6$. From Figure $6$, we see that entropy
measures of $H_{\beta_{1}}$, $H_{\beta_{2}}$, $H_{\beta_{3}}$, and
$H_{\beta_{8}}$ almost coincide.

Table 11: Numerical values of index-entropies of $J$ [$l,m$] | $H_{\beta_{1}}(J)$ | $H_{\beta_{2}}(J)$ | $H_{\beta_{3}}(J)$ | $H_{\beta_{4}}(J)$ | $H_{\beta_{5}}(J)$ | $H_{\beta_{6}}(J)$ | $H_{\beta_{7}}(J)$ | $H_{\beta_{8}}(J)$
---|---|---|---|---|---|---|---|---
[2,2] | 2.363878 | 2.349537 | 2.351098 | 2.293045 | 2.469266 | 1.849911 | 2.235934 | 2.358964
[3,3] | 2.680031 | 2.668041 | 2.669162 | 2.614962 | 2.751708 | 2.162725 | 2.567487 | 2.674998
[4,4] | 2.912369 | 2.901799 | 2.902659 | 2.851695 | 2.966026 | 2.393078 | 2.813031 | 2.907277
[5,5] | 3.09586 | 3.086229 | 3.086919 | 3.038506 | 3.138274 | 2.575276 | 3.00722 | 3.090734
[6,6] | 3.24743 | 3.238462 | 3.239031 | 3.192634 | 3.28218 | 2.725919 | 3.167437 | 3.242282
[7,7] | 3.37652 | 3.368045 | 3.368525 | 3.323745 | 3.40572 | 2.85433 | 3.303596 | 3.371357
[8,8] | 3.488993 | 3.480834 | 3.481247 | 3.437785 | 3.51397 | 2.966216 | 3.421864 | 3.483755
[9.9] | 3.588472 | 3.580678 | 3.581037 | 3.538669 | 3.610178 | 3.065343 | 3.526328 | 3.583289
[10,10] | 3.677788 | 3.670239 | 3.670554 | 3.629107 | 3.696846 | 3.154321 | 3.61983 | 3.672599
Figure 6: Graphical comparison among index-entropies of J

The values of index-entropy of Y-junction graph $J_{1}$ is listed in Table
$12$ and illustrated in Figure $7$. From Table $12$ and Figure $7$, we find
that measures of graph index-entropies $H_{\beta_{1}}$, $H_{\beta_{2}}$,
$H_{\beta_{3}}$, $H_{\beta_{5}}$, $H_{\beta_{6}}$, and $H_{\beta_{1}}$ are
almost same.

Table 12: Numerical values of index-entropies of $J_{1}$ [$l,m$] | $H_{\beta_{1}}(J_{1})$ | $H_{\beta_{2}}(J_{1})$ | $H_{\beta_{3}}(J_{1})$ | $H_{\beta_{4}}(J_{1})$ | $H_{\beta_{5}}(J_{1})$ | $H_{\beta_{6}}(J_{1})$ | $H_{\beta_{7}}(J_{1})$ | $H_{\beta_{8}}(J_{1})$
---|---|---|---|---|---|---|---|---
[2,2] | 2.374116 | 2.362875 | 2.365677 | 2.37483 | 2.351939 | 2.381171 | 2.336108 | 2.373399
[3,3] | 2.686117 | 2.677718 | 2.679751 | 2.705906 | 2.66972 | 2.691361 | 2.643717 | 2.686081
[4,4] | 2.916047 | 2.909359 | 2.91094 | 2.948613 | 2.903076 | 2.92019 | 2.871061 | 2.916411
[5,5] | 3.097981 | 3.092424 | 3.093709 | 3.139639 | 3.087253 | 3.101393 | 3.051307 | 3.098605
[6,6] | 3.248463 | 3.243705 | 3.244784 | 3.2969 | 3.239312 | 3.251355 | 3.200605 | 3.24927
[7,7] | 3.376753 | 3.372588 | 3.373514 | 3.430435 | 3.368768 | 3.379258 | 3.23802 | 3.377694
[8,8] | 3.488549 | 3.484842 | 3.485651 | 3.546395 | 3.481462 | 3.490754 | 3.439143 | 3.489594
[9.9] | 3.587606 | 3.584263 | 3.584979 | 3.648847 | 3.58132 | 3.589572 | 3.537668 | 3.588733
[10,10] | 3.676529 | 3.673482 | 3.674123 | 3.740586 | 3.6707383 | 3.6783 | 3.626158 | 3.677722
Figure 7: Graphical comparison among index-entropies of $J_{1}$

Table $13$ depicts some graph index-entropies of Y-junction graph $J_{2}$. The
graphical comparison of index-entropies of Y-junction graph $J_{2}$ is shown
in Figure $8$. From Figure $8$, we see that graph index-entropies of $J_{2}$
increases as the values of $l$ and $m$ increases.

Table 13: Numerical values of index-entropies of $J_{2}$ [$l,m$] | $H_{\beta_{1}}(J_{2})$ | $H_{\beta_{2}}(J_{2})$ | $H_{\beta_{3}}(J_{2})$ | $H_{\beta_{4}}(J_{2})$ | $H_{\beta_{5}}(J_{2})$ | $H_{\beta_{6}}(J_{2})$ | $H_{\beta_{7}}(J_{2})$ | $H_{\beta_{8}}(J_{2})$
---|---|---|---|---|---|---|---|---
[2,2] | 2.388128 | 2.375827 | 2.346955 | 1.633105 | 2.336917 | 2.391354 | 2.345766 | 2.35432
[3,3] | 2.696411 | 2.687189 | 2.666479 | 1.88376 | 2.665889 | 2.698832 | 2.650775 | 2.672367
[4,4] | 2.924151 | 2.916793 | 2.9007 | 2.074455 | 2.902766 | 2.926068 | 2.876596 | 2.905747
[5,5] | 3.104655 | 3.098533 | 3.085384 | 2.229346 | 3.088295 | 3.106231 | 3.055853 | 3.089892
[6,6] | 3.254134 | 3.248888 | 3.237774 | 2.360107 | 3.240916 | 3.255465 | 3.204459 | 3.241908
[7,7] | 3.381681 | 3.377087 | 3.367462 | 2.473394 | 3.370602 | 3.3882828 | 3.331363 | 3.371323
[8,8] | 3.492905 | 3.488816 | 3.480327 | 2.573399 | 3.48337 | 3.49391 | 3.442095 | 3.483978
[9.9] | 3.59151 | 3.587821 | 3.580028 | 2.662948 | 3.583139 | 3.592399 | 3.540309 | 3.583714
[10,10] | 3.680065 | 3.676703 | 3.659833 | 2.744042 | 3.672602 | 3.680861 | 3.628548 | 3.673185
Figure 8: Graphical comparison among index-entropies of $J_{2}$

In Table $14$, we calculate some graph index-entropies of Y-junction graph
$J_{3}$. Figure $9$ shows the graphical comparison among index-entropies of
$J_{3}$. From Table $14$ and Figure $9$, we see that index entropies
$H_{\beta_{1}}$, $H_{\beta_{2}}$, $H_{\beta_{3}}$, $H_{\beta_{6}}$, and
$H_{\beta_{8}}$ of $J_{3}$ are almost same. Also, Tables $11$, $12$, $13$, and
$14$ shows that graph index-entropies of Y-junction graph increases as the
values of $l$ and $m$ increases.

Table 14: Numerical values of index-entropies of $J_{3}$ [$l,m$] | $H_{\beta_{1}}(J_{3})$ | $H_{\beta_{2}}(J_{3})$ | $H_{\beta_{3}}(J_{3})$ | $H_{\beta_{4}}(J_{3})$ | $H_{\beta_{5}}(J_{3})$ | $H_{\beta_{6}}(J_{3})$ | $H_{\beta_{7}}(J_{3})$ | $H_{\beta_{8}}(J_{3})$
---|---|---|---|---|---|---|---|---
[2,2] | 2.401692 | 2.388393 | 2.393488 | 2.179494 | 1.755856 | 2.404539 | 2.359174 | 2.40079
[3,3] | 2.706459 | 2.696449 | 2.7002 | 2.469933 | 2.242514 | 2.708584 | 2.660759 | 2.70624
[4,4] | 2.932101 | 2.924095 | 2.927043 | 2.687 | 2.571439 | 2.933776 | 2.884517 | 2.932298
[5,5] | 3.11224 | 3.104551 | 3.106972 | 2.86049 | 2.816575 | 3.112596 | 3.062409 | 3.111698
[6,6] | 3.259727 | 3.254003 | 3.256051 | 3.005032 | 3.010781 | 3.260882 | 3.210048 | 3.260399
[7,7] | 3.38655 | 3.381534 | 3.383306 | 3.128925 | 3.171078 | 3.387542 | 3.336232 | 3.387369
[8,8] | 3.497215 | 3.492748 | 3.494307 | 3.237341 | 3.307302 | 3.498082 | 3.446407 | 3.498149
[9.9] | 3.595375 | 3.591345 | 3.592735 | 3.33372 | 3.425608 | 3.596141 | 3.544179 | 3.5964
[10,10] | 3.683569 | 3.679896 | 3.681147 | 3.420469 | 3.530087 | 3.684252 | 3.632058 | 3.684668
Figure 9: Graphical comparison among index-entropies of $J_{3}$

##  8 Conclusion and Future work

In this study, the general expression of NM-polynomial for carbon nanotube
Y-junction graphs is derived. Also, various neighborhood degree sum-based
topological indices are retrieved from the expression of these polynomials. In
addition, eight graph entropies in terms of these topological indices have
been defined and calculated for Y-junction graphs. Furthermore, some numerical
values of topological indices and index-entropies of Y-junction graphs are
plotted for comparison. Since topological indices based on the degree of
vertices has a significant ability to predict various physicochemical
properties and biological activities of the chemical molecule. Therefore, the
study’s findings will be a viable option for predicting various
physicochemical properties and understanding the structural problems of carbon
nanotube Y-junctions.

We mention some possible directions for future research, including
multiplicative topological indices, graph index-entropies, regression models
between the index-entropies and the topological indices, metric and edge
metric dimension, etc., to predict thermochemical data, physicochemical
properties, and structural information of carbon nanotube Y-junctions.

Data Availability
No data was used to support the findings of this study.

Conflicts of Interest
There are no conflicts of interest declared by the authors.

Funding Statement
The authors received no specific funding for this study.

Author’s Contribution Statement
The final draft was written by Sohan Lal and Vijay Kumar Bhat. Figures and
Tables are prepared by Sohan Lal and Sahil Sharma. All authors reviewed and
edited the final draft.

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