Analytical and computational properties of the variable symmetric division deg index

TL;DR

This paper derives new inequalities for the variable symmetric division degree index, characterizes extremal graphs, and demonstrates that the index's average ratio depends solely on the graph's average degree.

Contribution

It introduces generalized inequalities for the $SDD_eta(G)$ index and characterizes extremal graphs, extending previous results and analyzing its behavior on random graphs.

Findings

New inequalities for the $SDD_eta(G)$ index are established.

Extremal graphs for the inequalities are characterized.

The average $SDD_eta(G)$ ratio depends only on the graph's average degree.

Abstract

The aim of this work is to obtain new inequalities for the variable symmetric division deg index $SDD_\alpha(G) = \sum_{uv \in E(G)} (d_u^\alpha/d_v^\alpha+d_v^\alpha/d_u^\alpha)$, and to characterize graphs extremal with respect to them. Here, $uv$ denotes the edge of the graph $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $\alpha \in \mathbb{R}$. Some of these inequalities generalize and improve previous results for the symmetric division deg index. In addition, we computationally apply the $SDD_\alpha(G)$ index on random graphs and show that the ratio $\left\langle SDD_\alpha(G) \right\rangle/n$ ($n$ being the order of the graph) depends only on the average degree $\left\langle d \right\rangle$.