New improved lower bounds for Zagreb indices of graphs

TL;DR

This paper introduces new lower bounds for Zagreb indices of graphs, improving existing inequalities and providing applications to spectral radius, coindices, and related graph invariants.

Contribution

It offers sharper bounds for Zagreb indices at specific parameters and extends these results to various graph invariants and inequalities.

Findings

Sharper bounds for $M_1(G)$ and ${}^{m}M_1(G)$ at $ ext{α} = 2$ and $-2$

New bounds for second Zagreb index and spectral radius

Enhanced inequalities for Nordhaus-Gaddum type bounds

Abstract

This paper presents new lower bounds for the first general Zagreb index $Z_{\alpha}(G)$ involving two, three, and four arbitrary degrees of vertices of a simple graph $G$. For the special cases $\alpha = 2$ and $\alpha = -2$, the results give sharper bounds for the first Zagreb index $M_1(G)$ and the modified first Zagreb index ${}^{m}M_1(G)$, thereby improving several well-known inequalities in the literature. Furthermore, some applications of the derived bounds for $M_1(G)$ are demonstrated, establishing new bounds for the second Zagreb index, the spectral radius, Nordhaus-Gaddum type bounds, and their corresponding coindices.