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On the inverse sum indeg index of some graph operations


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.


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

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

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

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

where du 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 [1215].

Preparation of the manuscript

Throughout this paper, we assume that Gi=(Vi,Ei) where \( V_{i}\cap V_{j}=\varnothing \) and \(E_{i}\cap E_{j}=\varnothing, i\neq j\) with |Vi|=ni,|Ei|=mifori=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,...,xn be positive real numbers.

i The arithmetic mean of x 1,x2,...,xn 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,...,xn 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,...,xn be positive real numbers. Then, HM(x1,x2,...,xn)≤AM(x1,x2,...,xn) with equality if only if x 1=x2=...=xn.

Definition 2

Let x be a vertex xV(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:uV(G1)}[16]

Definition 3

The join G= G1+G2ofG1andG2is defined asG=(V,E)withV=V1V2 and where E=E1E2E, 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,...,Gk, where Gi=(Vi,Ei) 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+...+Gn=(V,E), where V=V1V2V3...Vk and where E=E1E2...EkE, is (G1+G2)(G2+G3)...(Gk−1+Gk)[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 v1v2E2 or v1=v2 and u1u2E1[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 u1u2E1 or u1=u2 and v1v2E2[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 G1G2, 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

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


We obtain

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

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

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

Let ΔG be the maximum degree of G. Then,

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

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

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

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

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


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

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,

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


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

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

From Theorem 1, we have

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


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

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

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


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

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++Gk. Then,

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


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

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

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,

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


Assume that ui,ukV(G1),vj,vlV(G2). From Definition 3, we can write

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


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

By using Theorem 1, we get

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

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

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

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

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

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,

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


Assume that ui,ukV(G1),vj,vlV(G2). From Definition 4 and \(d_{G_{1}[G_{2}]}=n_{2}d_{G_{1}}(u)+d_{G_{2}}(v),\) we get

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

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

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


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

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=G1G2. Then,

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


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

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


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


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

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

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


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.

Availability of data and materials

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.


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Havare, Ö.Ç. On the inverse sum indeg index of some graph operations. J Egypt Math Soc 28, 28 (2020).

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