Chebyshev's Sum Inequality and the Zagreb Indices Inequality

TL;DR

This paper revisits the Zagreb indices inequality in graph theory, showing it is not entirely new and extending similar inequalities to directed graphs, sum-symmetric matrices, and Eulerian graphs.

Contribution

The paper clarifies the novelty of the Zagreb indices inequality and extends related inequalities to broader classes of graphs and matrices.

Findings

The Zagreb indices inequality is not a new result.

Extensions to directed graphs and Eulerian graphs are discussed.

Related inequalities for sum-symmetric matrices are presented.

Abstract

In a recent article, Nadeem and Siddique used Chebyshev's sum inequality to establish the Zagreb indices inequality $M_1/n\le M_2/m$ for undirected graphs in the case where the degree sequence $(d_i)$ and the degree-sum sequence $(S_i)$ are similarly ordered. We show that this is actually not a completely new result and we discuss several related results that also cover similar inequalities for directed graphs, as well as sum-symmetric matrices and Eulerian directed graphs.