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 ∑dudvdu+dv\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sum \frac {d_{u}d_{v}}{d_{u}+d_{v}}$\end{document}. 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 structureproperty 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 ( 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 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][13][14][15].

Preparation of the manuscript
Throughout this paper, we assume that Lemma 1 [9] Let G be a graph of size m. Then, The arithmetic mean of x 1 , x 2 , ..., x n is equal to ii The harmonic mean of x 1 , x 2 , ..., x n is equal to Theorem 1 Let x 1 , x 2 , ..., x n be positive real numbers. Then,

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 G 1 and G 2 , denoted by G 1 •G 2 , is the graph obtained by taking one copy of G 1 of order n 1 and n 1 copies of G 2 , and then joining by an edge the ith vertex of G 1 to every vertex in the ith copy of G 2 . 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.
Proof Assume that u i , u k ∈ V (G 1 ), v j , v l ∈ V (G 2 ). From Definition 3, we can write By using Theorem 1, we get