Analytical and computational study of the variable inverse sum deg index

TL;DR

This paper investigates the variable inverse sum degree index in graphs, deriving new inequalities, characterizing extremal graphs, and validating findings through computational experiments on random graphs.

Contribution

It introduces new inequalities for the variable inverse sum deg index, generalizes previous results, and explores the index's behavior through computational validation.

Findings

Derived new inequalities for ISD_a index.

Generalized and improved existing bounds.

Validated inequalities on random graph ensembles.

Abstract

A large number of graph invariants of the form $\sum_{uv \in E(G)} F(d_u,d_v)$ are studied in mathematical chemistry, where $uv$ denotes the edge of the graph $G$ connecting the vertices $u$ and $v$, and $d_u$ is the degree of the vertex $u$. Among them the variable inverse sum deg index $ISD_a$, with $F(d_u,d_v)=1/(d_u^a+d_v^a)$, was found to have applicative properties. The aim of this paper is to obtain new inequalities for the variable inverse sum deg index, and to characterize graphs extremal with respect to them. Some of these inequalities generalize and improve previous results for the inverse sum deg index. In addition, we computationally validate some of the obtained inequalities on ensembles of random graphs and show that the ratio $\left\langle ISD_a(G) \right\rangle/n$ ($n$ being the order of the graph) depends only on the average degree $\left\langle d \right\rangle$.