Bounds on the Inverse symmetric division deg index and the relation with other topological indices of graphs

TL;DR

This paper establishes bounds for the inverse symmetric division degree index (ISDD) of graphs, explores its relationships with other topological indices, and identifies extremal graphs, contributing to the mathematical understanding of this recently introduced index.

Contribution

It provides new bounds for ISDD, relates it to other indices, and identifies extremal graphs, advancing the theoretical analysis of this novel index.

Findings

Derived lower and upper bounds for ISDD

Established relations between ISDD and other topological indices

Identified extremal graphs for ISDD

Abstract

Let $G=(V,E)$ be a simple graph. The concept of Inverse symmetric division deg index $(ISDD)$ was introduced in the chemical graph theory very recently. In spite of this, a few papers have already appeared with this index in the literature. Ghorbani et al. proposed Inverse symmetric division deg index and is defined as $$ISDD(G)=\sum\limits_{v_iv_j\in E(G)}\,\displaystyle{\frac{d_id_j}{d^2_i+d^2_j}},$$ where $d_i$ is the degree of the vertex $v_i$ in $G$. In this paper, we obtain some lower and upper bounds on the inverse symmetric division deg index $(ISDD)$ of graphs in terms of various graph parameters, with identifying extremal graphs. Moreover, we present two relations between the Inverse symmetric division deg index and the various topological indices of graphs. Finally, we give concluding remarks with future work.