New Lower Bounds for the First Variable Zagreb Index

TL;DR

This paper establishes new sharp lower bounds for the first variable Zagreb index and related topological indices, characterizing extremal graphs based on degree bounds, advancing graph theory bounds.

Contribution

It introduces novel lower bounds for a broad family of topological indices, including the first variable Zagreb index, using only degree extremities, and characterizes extremal graphs.

Findings

New sharp inequalities for topological indices

Lower bounds involving minimum and maximum degree

Characterization of extremal graphs

Abstract

The aim of this paper is to obtain new sharp inequalities for a large family of topological indices, including the first variable Zagreb index $M_1^\alpha$, and to characterize the set of extremal graphs with respect to them. Our main results provide lower bounds on this family of topological indices involving just the minimum and the maximum degree of the graph. These inequalities are new even for the first Zagreb, the inverse and the forgotten indices.