On the inverse sum indeg index of some graph operations

Abstract

Topological indices are the molecular descriptors that describe the structures of chemical compounds. They are used in isomer discrimination, structure-property relationship, and structure-activity relations. The topological indices are used to predict certain physico-chemical properties such as boiling point, enthalpy of vaporization, and stability. In this paper, the inverse sum indeg index is studied. This index (ISI(G)) is defined as $\sum \frac {d_{u}d_{v}}{d_{u}+d_{v}}$. The inverse sum indeg index of some graph operations is computed. These operations are join, sequential join, cartesian product, lexicographic product, and corona operation.

Introduction

A graph G is a finite nonempty vertex set V(G) together with a edge set E. An edge of G which is e connects the vertices u and v. It writes e=uv, says u and v are adjacent. We often use n and m for the order and the size of a graph, respectively [1].

Chemical graph theory is concerned with finding topological indices that are well correlated with the properties of chemical molecules. The edges and the vertices of a graph represent the bonds and the atoms of a molecule, respectively [2].

The topological index which is known as a graph-based molecular descriptor or graph invariant is the real values of the topological structure of a molecule [3].

Topological indices are used for studying the properties of molecules such as structure-property relationship (QSPR), structure-activity relationship (QSAR), and structural design in chemistry, nanotechnology, and pharmacology. Its main role is to work as a numerical molecular descriptor in QSAR/QSPR models [4, 5].

The first topological index is the Wiener index. In 1947, Harold Wiener introduced this index which was used to determine physical properties of paraffin [6]. It was used for the correlation of measured properties of molecules with their structural features by H. Wiener.

Many topological indices were defined. The Zagreb index is the most studied index. The first Zagreb index [7] was defined by Gutman and Trinajstić as

$$ M_{1}(G)=\underset{u\in V(G)}{\sum }d_{u}=\underset{uv\in E\left(G\right) }{ \sum }d_{u}+d_{v}. $$

(1)

In 2010, D. Vukicevic and M. Gasperov introduced adriatic indices that are obtained by the analyses of well-known indices such as the Randic and the Wiener index. D. Vukicevic and M. Gasperov performed QSAR and QSPR studies of adriatic indices [8]. Three classes of adriatic descriptors are defined. One of these descriptors is the discrete adriatic descriptors which consist of 148 descriptors. These descriptors have very good predictive properties. Thus, many scientists studied these indices. The inverse sum indeg index is one of the discrete adriatic descriptors. The inverse sum indeg index is defined as

$$ ISI(G)=\underset{uv\in E(G)}{\sum}\frac{1}{\frac{1}{d_{u}}+\frac{1}{d_{v}}}= \underset{uv\in E(G)}{\sum }\frac{d_{u}d_{v}}{d_{u}+d_{v}}, $$

(2)

where d_u is denoted as the degree of vertex u [8].

The inverse sum indeg index gives a significant predictor of total surface area of octane isomers. Nezhad et al. studied several sharp upper and lower bounds on the inverse sum indeg index [9]. Nezdah et al. computed the inverse sum indeg index of some nanotubes [10]. Sedlar et al. presented extremal values of this index across several graph classes such as trees and chemical trees [11]. Many scientists studied the topological index of graph operations. We encourage to examine the references that are given here [12–15].

Preparation of the manuscript

Throughout this paper, we assume that G_i=(V_i,E_i) where $V_{i}\cap V_{j}=\varnothing$ and $E_{i}\cap E_{j}=\varnothing, i\neq j$ with |V_i|=n_i,|E_i|=m_ifori=1,2,...,k.

Lemma 1

[9] Let G be a graph of size m. Then,

$$\underset{u\in V(G)}{\sum }d_{u}=2m. $$

Definition 1

Let x ₁,x₂,...,x_n be positive real numbers.

i The arithmetic mean of x ₁,x₂,...,x_n is equal to

$$AM(x_{1},x_{2},...,x_{n})=\frac{x_{1}+x_{2}+...+x_{n}}{n}. $$

ii The harmonic mean of x ₁,x₂,...,x_n is equal to

$$HM(x_{1},x_{2},...,x_{n})=\frac{n}{\frac{1}{x_{1}}+\frac{1}{x_{2}}+...+\frac{1}{x_{n}}}. $$

Theorem 1

Let x₁,x₂,...,x_n be positive real numbers. Then, HM(x₁,x₂,...,x_n)≤AM(x₁,x₂,...,x_n) with equality if only if x ₁=x₂=...=x_n.

Definition 2

Let x be a vertex x∉V(G₁). Then, G₁+{x} is a graph that is obtained from G₁ by including the vertex x and joining it to all other vertices of G₁. That is, G₁+{x}=(V,E), where V=V(G₁)∪{x} and E=E(G₁)∪{ux:u∈V(G₁)}[16]

Definition 3

The join G= G₁+G₂ofG₁andG₂is defined asG=(V,E)withV=V₁∪V₂ and where E=E₁∪E₂∪E^′, where E^′ is the set of all edges joining vertices of V₁ with vertices of V₂[17].

Definition 4

For three or more disjoint graphs, G₁,G₂,G₃,...,G_k, where G_i=(V_i,E_i) and where $V_{i}\cap V_{j}=\varnothing$ and $E_{i}\cap E_{j}=\varnothing, i\neq j,1\leq i,j\leq k$the sequential joinG=G₁+G₂+G₃+...+G_n=(V,E), where V=V₁∪V₂∪V₃∪...∪V_k and where E=E₁∪E₂∪...∪E_k∪E^′, is (G₁+G₂)∪(G₂+G₃)∪...∪(G_k−1+G_k)[17].

Definition 5

The cartesian product of G₁ and G₂, denoted G₁×G₂=(V,E), is a graph having V=V₁×V₂ and two vertices (u₁,v₁) and (u₂,v₂) are adjacent if only if either u₁=u₂ and v₁v₂∈E₂ or v₁=v₂ and u₁u₂∈E₁[17].

Definition 6

The composition known as lexicographic product G=G₁[G₂] of graphs G₁ and G₂ is the graph with vertex set V=V₁×V₂ and any two vertices (u₁,v₁) and (u₂,v₂) are adjacent if only if u₁u₂∈E₁ or u₁=u₂ and v₁v₂∈E₂[16].

Definition 7

The corona of two graphs was defined in [16], and there have been some results on the corona of two graphs [12]. The corona product of two graphs G₁ and G₂, denoted by G₁∘G₂, is the graph obtained by taking one copy of G₁ of order n₁ and n₁ copies of G₂, and then joining by an edge the ith vertex of G₁ to every vertex in the ith copy of G₂. The corona product is neither associative nor commutative.

Main results

In this section, it is given sharp bounds on the inverse sum indeg index of above graph operations.

Theorem 2

Let G=G₁+{x}, means that add a new vertex to the graph G₁. For ISI(G), the following holds

$$ ISI(G)\leq \frac{1}{4}M_{1}(G)+\frac{m_{1}}{2}+\frac{n_{1}^{2}}{2}. $$

Proof

We obtain

$$\begin{array}{@{}rcl@{}} ISI(G)&=&\underset{uv\in E(G)}{\sum }\frac{d_{u}d_{v}}{d_{u}+d_{v}} \notag \\ && =\underset{uv\in E(G_{1})}{\sum}\frac{(d_{u}+1)(d_{v}+1)}{d_{u}+1+d_{v}+1}+\underset{uv\in E^{^{\prime }}}{\sum }\frac{(d_{u}+1)n_{1}}{d_{u}+1+n_{1}}, \end{array} $$

(3)

where $\ E^{^{\prime }}$ is the set of all edges joining vertices of V₁ with x vertex. By using Theorem 1, we have

$$ \frac{(d_{u}+1)(d_{v}+1)}{d_{u}+1+d_{v}+1}=\frac{1}{2}\frac{2}{\frac{1}{ d_{u}+1}+\frac{1}{d_{v}+1}}\leq \frac{1}{2}\left(\frac{d_{u}+d_{v}}{2} +1\right). $$

(4)

Let Δ_G be the maximum degree of G. Then,

$$ \frac{(d_{u}+1)n_{1}}{d_{u}+1+n_{1}}\leq \frac{\Delta_{G}n_{1}}{\Delta _{G}+n_{1}}. $$

(5)

Note that Δ_G=n₁. So, Eq. (5) can be rewritten as

$$ \frac{(d_{u}+1)n_{1}}{d_{u}+1+n_{1}}\leq \frac{\Delta_{G}n_{1}}{\Delta _{G}+n_{1}}=\frac{n_{1}}{2}. $$

(6)

From Eqs. (3), (4), and (6), we have

$$\begin{array}{@{}rcl@{}} ISI(G)&=&\underset{uv\in E(G_{1})}{\sum }\frac{(d_{u}+1)(d_{v}+1)}{ d_{u}+1+d_{v}+1}+\underset{uv\in E^{^{\prime }}}{\sum }\frac{(d_{u}+1)n_{1}}{ d_{u}+1+n_{1}} \\ &&\leq \frac{1}{2}\underset{uv\in E(G_{1})}{\sum }\left(\frac{ d_{u}+d_{v}}{2}+1\right) +\frac{1}{2}\underset{uv\in E^{^{\prime }}}{\sum } n_{1} \end{array} $$

or

$$ ISI(G)\leq \frac{1}{4}\underset{uv\in E(G_{1})}{\sum }\left(d_{u}+d_{v}\right) +\frac{1}{2}\underset{uv\in E(G_{1})}{\sum }1+\frac{n_{1} }{2}\underset{uv\in E^{^{\prime }}}{\sum }1. $$

From Eq. (1), we can write

$$ ISI(G)\leq \frac{1}{4}M_{1}(G)+\frac{m_{1}}{2}+\frac{n_{1}}{2}n_{1}. $$

□

Theorem 3

Let G=G₁+G₂. Then,

$$\begin{array}{@{}rcl@{}} ISI(G)&\leq & \frac{1}{4}\left(M_{1}(G_{1})+M_{1}(G_{2})\right) +\frac{ m_{1}+m_{2}}{2}+n_{1}n_{2}\left(\frac{n_{1}+n_{2}}{4}\right) + \\ && \frac{ m_{2}n_{1}+m_{1}n_{2}}{2}. \end{array} $$

Proof

From Eq. (2) and Definition 3, we have

$$\begin{array}{@{}rcl@{}} ISI(G)&=&\underset{uv\in E(G_{1})}{\sum }\frac{(d_{u}+n_{2})(d_{v}+n_{2})}{ d_{u}+n_{2}+d_{v}+n_{2}}+\underset{uv\in E(G_{2})}{\sum }\frac{ (d_{u}+n_{1})(d_{v}+n_{1})}{d_{u}+n_{1}+d_{v}+n_{1}}+ \notag \\ && \underset{v\in V(G_{1})} {\underset{u\in V(G_{2})}{\underset{uv\in E^{^{\prime }}}{\sum }}}\frac{ (d_{u}+n_{2})\left(d_{v}+n_{1}\right) }{d_{u}+n_{2}+d_{v}+n_{1}}. \end{array} $$

(7)

From Theorem 1, we have

$$ \frac{(d_{u}+n_{2})(d_{v}+n_{2})}{d_{u}+n_{2}+d_{v}+n_{2}}=\frac{1}{2}\frac{2 }{\frac{1}{d_{u}+n_{2}}+\frac{1}{d_{v}+n_{2}}}\leq \frac{1}{2}\left(\frac{ d_{u}+d_{v}}{2}+n_{2}\right), $$

(8)

$$ \frac{(d_{u}+n_{1})(d_{v}+n_{1})}{d_{u}+n_{1}+d_{v}+n_{1}}=\frac{1}{2}\frac{2 }{\frac{1}{d_{u}+n_{1}}+\frac{1}{d_{v}+n_{1}}}\leq \frac{1}{2}\left(\frac{ d_{u}+d_{v}}{2}+n_{1}\right) $$

(9)

and

$$ \frac{(d_{u}+n_{1})(d_{v}+n_{2})}{d_{u}+n_{1}+d_{v}+n_{2}}=\frac{1}{2}\frac{2 }{\frac{1}{d_{u}+n_{1}}+\frac{1}{d_{v}+n_{2}}}\leq \frac{1}{2}\left(\frac{ d_{u}+d_{v}}{2}+\frac{n_{2}+n_{1}}{2}\right). $$

(10)

Equation (7) can be rewritten with Eqs. (8), (9), and (10) as

$$\begin{array}{@{}rcl@{}} ISI(G)&\leq& \frac{1}{2}\underset{uv\in E(G_{1})}{\sum }\left(\frac{ d_{u}+d_{v}}{2}+n_{2}\right) +\frac{1}{2}\underset{uv\in E(G_{2})}{\sum } \left(\frac{d_{u}+d_{v}}{2}+n_{1}\right) + \\ &&\frac{1}{2}\underset{v\in V(G_{1}) }{\underset{u\in V(G_{2})}{\underset{uv\in E^{^{\prime }}}{\sum }}}\left(\frac{d_{u}+d_{v}}{2}+\frac{n_{1}+n_{2}}{2}\right). \end{array} $$

By using Eq. (1), we get

$$\begin{array}{@{}rcl@{}} ISI(G)&\leq& \frac{1}{4}M_{1}(G_{1})+\frac{n_{2}}{2}\underset{uv\in E(G_{1})}{ \sum }1+\frac{1}{4}M_{1}(G_{2})+\frac{n_{1}}{2}\underset{uv\in E(G_{2})}{ \sum }1+ \\ && \frac{1}{2}\underset{uv\in E^{^{\prime }}}{\sum }\left(\frac{ d_{u}+d_{v}}{2}+\frac{n_{1}+n_{2}}{2}\right) \end{array} $$

or

$$\begin{array}{@{}rcl@{}} ISI(G)&\leq& \frac{1}{4}M_{1}(G_{1})+\frac{n_{2}}{2}m_{1}+\frac{1}{4} M_{1}(G_{2})+\frac{n_{1}}{2}m_{2}+\frac{1}{2}\underset{uv\in E^{^{\prime }}}{ \sum }\frac{n_{1}+n_{2}}{2}+ \\ && \frac{1}{4}\underset{v\in V(G_{1})}{\sum }d_{v}+ \frac{1}{4}\underset{u\in V(G_{2})}{\sum }d_{u}. \end{array} $$

By Lemma 1, we have

$$\begin{array}{@{}rcl@{}} ISI(G)&\leq& \frac{1}{4}M_{1}(G_{1})+\frac{n_{2}}{2}m_{1}+\frac{1}{4} M_{1}(G_{2})+\frac{n_{1}}{2}m_{2}+\frac{1}{2}\left(\frac{n_{1}+n_{2}}{2} \right) n_{1}n_{2}+ \\&& \frac{1}{4}2m_{1}+\frac{1}{4}2m_{2}. \end{array} $$

□

Theorem 4

Let G=G₁+G₂+⋯+G_k. Then,

$$\begin{array}{@{}rcl@{}} ISI(G) &\leq &\frac{1}{4}\underset{j=1}{\overset{k}{\sum }}M_{1}(G_{j})+ \frac{1}{2}\underset{j=2}{\overset{k}{\sum }}m_{j}n_{j-1}+\frac{1}{2} \underset{j=1}{\overset{k-1}{\sum }}m_{j}n_{j+1}+ \\ &&\frac{1}{4}\underset{j=1}{ \overset{k-1}{\sum }}\left(n_{j}^{2}n_{j+1}+n_{j+1}^{2}n_{j}\right) +\frac{1 }{2}\underset{j=1}{\overset{k-2}{\sum }}n_{j}n_{j+1}n_{j+2} \\ &&+\frac{1}{2}\underset{j=1}{\overset{k-1}{\sum }}{m_{j}+m_{j+1}}. \end{array} $$

Proof

From Eq. (2) and Definition 3, we have

$$\begin{array}{@{}rcl@{}} ISI(G)&=&\underset{uv\in E(G_{1})}{\sum }\frac{(d_{u}+n_{2})(d_{v}+n_{2})}{ (d_{u}+n_{2}+d_{v}+n_{2})}+ \notag \\ &&\underset{j=\overline{2,k-1}}{\underset{uv\in E(G_{j})}{\sum }}\frac{(d_{u}+n_{(j-1)}+n_{(j+1)})(d_{v}+n_{(j-1)}+n_{(j+1)}) }{d_{u}+n_{(j-1)}+n_{(j+1)}+d_{v}+n_{(j-1)}+n_{(j+1)}}+ \notag \\ &&\underset{uv\in E(G_{k})}{\sum }\frac{(d_{u}+n_{k-1})(d_{v}+n_{k-1})}{ d_{u}+n_{k-1}+d_{v}+n_{k-1}}+ \notag \\ &&\underset{v\in V(G_{2})}{\underset{u\in V(G_{1}) }{\underset{uv\in E^{^{\prime }}}{\sum }}}\frac{ (d_{u}+n_{2})(d_{v}+n_{1}+n_{3})}{d_{u}+n_{2}+d_{v}+n_{1}+n_{3}}+ \notag\\ &&\underset{j=\overline{2,k-1}}{\underset{v\in V(G_{j+1})}{\underset{u\in V(G_{j})}{\underset{uv\in E^{^{\prime }}}{\sum }}}}\frac{ (d_{u}+n_{j-1}+n_{j+1})(d_{v}+n_{j}+n_{j+2})}{d_{u}+n_{j-1}+n_{j+1}+d_{v}+n_{j}+n_{j+2}}+\notag \\ &&\underset{v\in V(G_{k})}{\underset{u\in V(G_{k-1})}{\underset{uv\in E^{^{\prime }}}{\sum }}}\frac{ (d_{u}+n_{k-2}+n_{k})(d_{v}+n_{k-1})}{d_{u}+n_{k-2}+n_{k}+d_{v}+n_{k-1}}. \end{array} $$

(11)

Equation (11) can be rewritten using Theorem 1:

$$\begin{array}{@{}rcl@{}} ISI(G) &=&\underset{uv\in E(G_{1})}{\sum }\frac{1}{4}(d_{u}+d_{v}+2n_{2})+ \underset{j=\overline{2,k-1}}{\underset{uv\in E(G_{j})}{\sum }}\frac{1}{4} (d_{u}+d_{v}+2n_{j-1}+2n_{j+1})+ \\ &&\underset{uv\in E(G_{k})}{\sum }\frac{1}{4}(d_{u}+d_{v}+2n_{k-1})+\underset {v\in V(G_{2})}{\underset{u\in V(G_{1})}{\underset{uv\in E^{^{\prime }}}{ \sum }}}\frac{1}{4}(d_{u}+d_{v}+n_{1}+n_{2}+n_{3})+ \\ &&\underset{j=\overline{2,k-1}}{\underset{v\in V(G_{j+1})}{\underset{u\in V(G_{j})}{\underset{uv\in E^{^{\prime }}}{\sum }}}}\frac{1}{4} (d_{u}+d_{v}+n_{j-1}+n_{j}+n_{j+1}+n_{j+2})+ \\ &&\underset{v\in V(G_{k})}{ \underset{u\in V(G_{k-1})}{\underset{uv\in E^{^{\prime }}}{\sum }}}\frac{1}{4 }(d_{u}+d_{v}+n_{k-2}+n_{k-1}+n_{k}). \end{array} $$

By using Eq. (1), we get

$$\begin{array}{@{}rcl@{}} ISI(G) &\leq &\frac{1}{4}\left(M_{1}(G_{1})+2n_{2}m_{1}\right) +\\ &&\frac{1}{4} \underset{2\leq j\leq k-1}{\sum }(M_{1}(G_{j})+m_{j}(2n_{j-1}+2n_{j+1}))+ \\ &&\frac{1}{4}\left(M_{1}(G_{k})+2n_{k-1}m_{k}\right) + \frac{1}{4}\left(\underset{u\in V(G_{1})}{\sum }d_{v}+\underset{v\in V(G_{2})}{\sum }d_{u}\right) + \\ &&\frac{n_{1}n_{2}}{4}\left(n_{1}+n_{2}+n_{3}\right) +\frac{1}{4}\left(\underset{j=\overline{2,k-2}}{ \underset{u\in V(G_{j})}{\sum }}d_{v}+\underset{j=\overline{2,k-2}}{\underset {v\in V(G_{j+1})}{\sum }}d_{u}\right) + \\ &&\frac{n_{j}n_{j+1}}{4}\left(n_{j-1}+n_{j}+n_{j+1}+n_{j+2}\right) +\frac{1 }{4} \underset{u\in V(G_{k-1})}{\sum }d_{v}+ \frac{1}{4}\underset{v\in V(G_{k})}{ \sum }d_{u} + \\ &&\frac{n_{k-1}n_{k}}{4}\left(n_{k-2}+n_{k-1}+n_{k}\right). \end{array} $$

By Lemma 1, the proof is completed as

$$\begin{array}{@{}rcl@{}} ISI(G) &\leq &\frac{1}{4}\left(M_{1}(G_{1})+2n_{2}m_{1}\right) +\frac{1}{4} \underset{2\leq j\leq k-1}{\sum }(M_{1}(G_{j})+m_{j}(2n_{j-1}+2n_{j+1}))+ \\ &&\frac{1}{4}\left(M_{1}(G_{k})+2n_{k-1}m_{k}\right) + \frac{1}{4}\left(2m_{1}+2m_{2}\right) +\frac{n_{1}n_{2}}{4}\left(n_{1}+n_{2}+n_{3}\right) + \\ &&\frac{1}{4}\underset{j=\overline{2,k-2}}{\sum } \left(2m_{j}+2m_{j+1}\right) + \frac{n_{j}n_{j+1}}{4}\left(n_{j-1}+n_{j}+n_{j+1}+n_{j+2}\right) + \\ &&\frac{1 }{4}\left(2m_{k-1}+2m_{k}\right) +\frac{n_{k-1}n_{k}}{4}\left(n_{k-2}+n_{k-1}+n_{k}\right). \end{array} $$

□

Theorem 5

Let G=G₁×G₂. Then,

$$ ISI(G)\leq \frac{1}{4}\left(n_{2}M_{1}(G_{1})+n_{1}M_{1}(G_{2})\right) + \frac{m_{1}n_{2}\Delta_{2}+m_{2}n_{1}\Delta_{1}}{2}. $$

Proof

Assume that u_i,u_k∈V(G₁),v_j,v_l∈V(G₂). From Definition 3, we can write

$$ ISI(G)=\underset{(u_{i},v_{j})(u_{k},v_{l})\in E(G)}{\sum }\frac{d_{u}d_{v}}{ d_{u}+d_{v}} $$

or

$$ISI(G)=\underset{j\neq l}{\underset{(u_{i},v_{j})(u_{i},v_{l})\in E(G)} {\sum }}\frac{1}{\frac{1}{d_{u_{i}}+d_{v_{j}}}+\frac{1}{d_{u_{i}}+d_{v_{l}}}}+\underset{(u_{i},v_{j})(u_{k},v_{j})\in E(G)}{\sum }\frac{1}{\frac{1}{ d_{u_{i}}+d_{v_{j}}}+\frac{1}{d_{u_{k}}+d_{vj}}}. $$

(12)

By using Theorem 1, we get

$$ \frac{1}{\frac{1}{d_{u_{i}}+d_{v_{j}}}+\frac{1}{d_{u_{i}}+d_{v_{l}}}}\leq \frac{1}{2}\frac{d_{u_{i}}+d_{v_{j}}+d_{u_{i}}+d_{v_{l}}}{2}=\frac{d_{u_{i}} }{2}+\frac{d_{u_{i}}+d_{v_{l}}}{4}, $$

(13)

$$ \frac{1}{\frac{1}{d_{u_{i}}+d_{v_{j}}}+\frac{1}{d_{u_{k}}+d_{vj}}}\leq\frac{1}{2 }\frac{d_{u_{i}}+d_{v_{j}}+d_{u_{k}}+d_{vj}}{2}=\frac{d_{v_{j}}}{2}+\frac{ d_{u_{i}}+d_{u_{k}}}{4}. $$

(14)

Equation (12) is rewritten using Eqs. (13) and (14).

$$\begin{array}{@{}rcl@{}} ISI(G)&\leq& \underset{u_{i}\in V(G_{1})}{\sum }\underset{(v_{j},v_{l})\in E(G_{2})}{\sum }\left(\frac{d_{u_{i}}}{2}+\frac{d_{u_{i}}+d_{v_{l}}}{4} \right)+ \notag\\ &&\underset{v_{j}\in V(G_{2})}{\sum }\underset{(u_{i},u_{k})\in E(G_{1})}{\sum }\left(\frac{d_{v_{j}}}{2}+\frac{d_{u_{i}}+d_{u_{k}}}{4} \right). \end{array} $$

(15)

Let Δ₁,Δ₂ be the maximum degree of G ₁,G₂ respectively.

$$ \frac{d_{u_{i}}}{2}+\frac{d_{u_{i}}+d_{v_{l}}}{4}\leq \frac{\Delta_{1}}{2}+ \frac{d_{u_{i}}+d_{v_{l}}}{4}, $$

(16)

$$ \frac{d_{v_{j}}}{2}+\frac{d_{u_{i}}+d_{u_{k}}}{4}\leq \frac{\Delta_{2}}{2}+ \frac{d_{u_{i}}+d_{u_{k}}}{4}. $$

(17)

By Eqs. (16) and (17), we have

$$\begin{array}{@{}rcl@{}} ISI(G)&\leq&\underset{u_{i}\in V(G_{1})}{\sum }\underset{(v_{j},v_{l})\in E(G_{2})}{\sum }\left(\frac{\Delta_{1}}{2}+\frac{d_{u_{i}}+d_{v_{l}}}{4} \right) + \\ &&\underset{v_{j}\in V(G_{2})}{\sum }\underset{(u_{i},u_{k})\in E(G)}{ \sum }\left(\frac{\Delta_{2}}{2}+\frac{d_{u_{i}}+d_{u_{k}}}{4}\right). \end{array} $$

From Eq. (1), we get

$$ ISI(G)\leq \underset{u_{i}\in V(G_{1})}{\sum }\left(\frac{\Delta_{1}}{2}+ \frac{M_{1}(G_{2})}{4}\right) +\underset{v_{j}\in V(G_{2})}{\sum }\left(\frac{\Delta_{2}}{2}+\frac{M_{1}(G_{1})}{4}\right). $$

The following is obtained:

$$ ISI(G)\leq \frac{m_{2}n_{1}\Delta_{1}}{2}+\frac{n_{1}M_{1}(G_{2})}{4}+\frac{ m_{1}n_{2}\Delta_{2}}{2}+\frac{n_{2}M_{1}(G_{1})}{4}. $$

□

Theorem 6

Let G=G₁[G₂]. Then,

$$ ISI(G)\leq n_{2}\Delta_{1}m_{2}+\Delta_{2}m_{1}+\frac{M_{1}(G_{2})}{2}+ \frac{n_{2}M_{1}(G_{1})}{2}. $$

Proof

Assume that u_i,u_k∈V(G₁),v_j,v_l∈V(G₂). From Definition 4 and $d_{G_{1}[G_{2}]}=n_{2}d_{G_{1}}(u)+d_{G_{2}}(v),$ we get

$$\begin{array}{@{}rcl@{}} ISI(G)&=&\underset{(u_{i},v_{j})(u_{k},v_{l})\in E(G)}{\sum }\frac{ d_{u}d_{v}}{d_{u}+d_{v}} \notag \\ &&=\underset{u_{i}\in V(G_{1})}{\sum }\underset{j\neq l }{\underset{(v_{j},v_{l})\in E(G_{2})}{\sum }}\frac{1}{\frac{1}{ n_{2}d_{u_{i}}+d_{v_{j}}}+\frac{1}{n_{2}d_{u_{i}}+d_{v_{l}}}} \notag \\ &&+\underset{u_{j}\in V(G_{2})}{\sum }\underset{v_{l}\in V(G_{2})}{\sum } \underset{(u_{i},u_{k})\in E(G_{1})}{\sum }\frac{1}{\frac{1}{ n_{2}d_{u_{i}}+d_{v_{j}}}+\frac{1}{n_{2}d_{u_{k}}+d_{v_{l}}}}. \end{array} $$

(18)

Assume that Δ₁,Δ₂ be the maximum degree of G₁,G₂ respectively. From Theorem 1, we have

$$ \frac{1}{2}\frac{2}{\frac{1}{n_{2}d_{u_{i}}+d_{v_{j}}}+\frac{1}{ n_{2}d_{u_{i}}+d_{v_{l}}}}\leq \frac{ n_{2}d_{u_{i}}+d_{v_{j}}+n_{2}d_{u_{i}}+d_{v_{l}}}{2}\leq n_{2}\Delta_{1}+ \frac{d_{v_{j}}+d_{v_{l}}}{2} $$

(19)

and

$$ \frac{1}{2}\frac{2}{\frac{1}{n_{2}d_{u_{i}}+d_{v_{j}}}+\frac{1}{ n_{2}d_{u_{k}}+d_{v_{l}}}}\leq \frac{ n_{2}d_{u_{i}}+d_{v_{j}}+n_{2}d_{u_{k}}+d_{v_{l}}}{2}\leq \Delta_{2}+\frac{ n_{2}(d_{u_{i}}+d_{u_{k}})}{2}. $$

(20)

Equation (18) is rewritten by Eqs. (19) and (20):

$$ ISI(G)\leq \underset{v_{j}v_{l}\in E(G_{2})}{\sum }\left(n_{2}\Delta_{1}+ \frac{d_{v_{j}}+d_{v_{l}}}{2}\right) +\underset{u_{i}u_{k}\in E(G_{2})}{\sum }\left(\Delta_{2}+\frac{n_{2}(d_{u_{i}}+d_{u_{k}})}{2}\right). $$

By Eq. (1), it is obtained as

$$ ISI(G)\leq n_{2}\Delta_{1}m_{2}+\frac{M_{1}(G_{2})}{2}+\frac{ n_{2}M_{1}(G_{1})}{2}+\Delta_{2}m_{1}. $$

□

Theorem 7

Let G=G₁∘G₂. Then,

$$\begin{array}{@{}rcl@{}} ISI(G)&\leq& \frac{\Delta_{1}}{\delta_{1}+n_{2}}ISI(G_{1})+\frac{n_{1}\Delta _{2}}{\delta_{2}+1}ISI(G_{2})+n_{2}m_{1}\frac{2\Delta_{1}+n_{2}}{2\delta _{1}+2n_{2}}+ \\ &&n_{1}m_{2}\frac{2\Delta_{2}+1}{2\delta_{2}+2}+\frac{\left(\Delta_{1}+n_{2}\right) \left(\Delta_{2}+1\right) }{\delta_{1}+\delta _{2}+n_{2}+1}n_{1}n_{2}. \end{array} $$

Proof

From Definition 7, we have

$$\begin{array}{@{}rcl@{}} ISI(G)&=&\underset{uv\in E(G_{1})}{\sum }\frac{(d_{u}+n_{2})(d_{v}+n_{2})}{ d_{u}+n_{2}+d_{v}+n_{2}}+n_{1}\underset{uv\in E(G_{2})}{\sum }\frac{ (d_{u}+1)(d_{v}+1)}{d_{u}+d_{v}+2}+ \\ &&\underset{v\in V(G_{1})}{\underset{u\in V(G_{2})}{\underset{uv\in E^{^{\prime }}}{\sum }}}\frac{ (d_{u}+n_{2})(d_{v}+1)}{d_{u}+n_{2}+d_{v}+1}. \end{array} $$

Note that

$$\begin{array}{@{}rcl@{}} \frac{(d_{u}+n_{2})(d_{v}+n_{2})}{d_{u}+n_{2}+d_{v}+n_{2}}&=&\frac{d_{u}d_{v}}{ d_{u}+d_{v}+2n_{2}}+\frac{n_{2}(d_{u}+d_{v})+n_{2}^{2}}{d_{u}+d_{v}+2n_{2}} \\ &&\leq \frac{d_{u}d_{v}}{d_{u}+d_{v}}\frac{d_{u}+d_{v}}{d_{u}+d_{v}+2n_{2}}+ \frac{n_{2}(d_{u}+d_{v})+n_{2}^{2}}{d_{u}+d_{v}+2n_{2}} \end{array} $$

and

$$\begin{array}{@{}rcl@{}} \frac{(d_{u}+1)(d_{v}+1)}{d_{u}+d_{v}+2}&=&\frac{d_{u}d_{v}}{d_{u}+d_{v}+2}+ \frac{d_{u}+d_{v}+1}{d_{u}+d_{v}+2}\\ &&\leq \frac{d_{u}d_{v}}{d_{u}+d_{v}}\frac{ d_{u}+d_{v}}{d_{u}+d_{v}+2}+\frac{d_{u}+d_{v}+1}{d_{u}+d_{v}+2}. \end{array} $$

Then,

$$\begin{array}{@{}rcl@{}} ISI(G) &\leq &\underset{uv\in E(G_{1})}{\sum }\frac{d_{u}d_{v}}{d_{u}+d_{v}} \frac{d_{u}+d_{v}}{d_{u}+d_{v}+2n_{2}}+\underset{uv\in E(G_{1})}{\sum }\frac{ n_{2}(d_{u}+d_{v})+n_{2}^{2}}{d_{u}+d_{v}+2n_{2}}+ \\ &&n_{1}\underset{uv\in E(G_{2})}{\sum }\frac{d_{u}d_{v}}{d_{u}+d_{v}}\frac{d_{u}+d_{v}}{ d_{u}+d_{v}+2}+n_{1}\underset{uv\in E(G_{2})}{\sum }\frac{d_{u}+d_{v}+1}{d_{u}+d_{v}+2}+ \\ && \underset{v\in V(G_{1})}{\underset{u\in V(G_{2})}{\underset{uv\in E^{^{\prime }}}{\sum }}}\frac{(d_{u}+n_{2})(d_{v}+1)}{d_{u}+n_{2}+d_{v}+1}. \end{array} $$

Assume that Δ₁(δ₁),Δ₂(δ₂) be maximum (minimum) degree of G₁,G₂ respectively.

$$\begin{array}{@{}rcl@{}} ISI(G) &\leq &\underset{uv\in E(G_{1})}{\sum }\frac{d_{u}d_{v}}{d_{u}+d_{v}} \frac{2\Delta_{1}}{2\delta_{1}+2n_{2}}+\underset{uv\in E(G_{1})}{\sum } \frac{n_{2}2\Delta_{1}+n_{2}^{2}}{2\delta_{1}+2n_{2}}+ \\ &&n_{1}\underset{uv\in E(G_{2})}{\sum}\frac{d_{u}d_{v}}{d_{u}+d_{v}}\frac{2\Delta_{2}}{2\delta _{2}+2}+n_{1}\underset{uv\in E(G_{2})}{\sum}\frac{2\Delta_{2}+1}{2\delta_{2}+2} + \\ &&\underset{v\in V(G_{1})}{\underset{u\in V(G_{2})}{\underset{uv\in E^{^{\prime }}}{\sum }}}\frac{(\Delta_{1}+n_{2})(\Delta_{2}+1)}{\delta _{1}+n_{2}+\delta_{2}+1}. \end{array} $$

By Eq. (2), we obtain

$$\begin{array}{@{}rcl@{}} ISI(G)&\leq& \frac{\Delta_{1}}{\delta_{1}+n_{2}}ISI(G_{1})+\frac{n_{1}\Delta _{2}}{\delta_{2}+1}ISI(G_{2})+n_{2}m_{1}\frac{2\Delta_{1}+n_{2}}{2\delta _{1}+2n_{2}}+ \\ &&n_{1}m_{2}\frac{2\Delta_{2}+1}{2\delta_{2}+2}+\frac{\left(\Delta_{1}+n_{2}\right) \left(\Delta_{2}+1\right) }{\delta_{1}+\delta _{2}+n_{2}+1}n_{1}n_{2}. \end{array} $$

□

Conclusions

The topological indices are used theoretically to predict the physical-chemical properties of a chemical structure. In particular, they are used to estimate the physical and chemical properties of the new molecular structure without experimentation.

The ISI(G) index which is a significant predictor of the total surface area of octane isomers has been many studied among topological indices. The graph operations play an important role in graph theory. Upper bounds for new graphs that are obtained by graph operations are given. These bounds are based on minimum-maximum degree, vertex-edge numbers. The results of this study may be used as a predictor especially in the chemical graph theory.