On the Discrepancy Between Two Zagreb Indices

TL;DR

This paper investigates the maximum value of a graph invariant related to Zagreb indices, providing exact results for various classes of graphs and edge counts, with implications for mathematical chemistry.

Contribution

It characterizes the maximum of the sum of minimum degrees over edges for different graph classes and edge counts, extending understanding of Zagreb indices.

Findings

Maximum S(G) achieved by three graph classes depending on edge count.

Exact maximum S(G) values for bipartite, forest, and planar graphs.

Provides bounds and constructions for graphs with given edge numbers.

Abstract

We examine the quantity \[S(G) = \sum_{uv\in E(G)} \min(\text{deg } u, \text{deg } v)\] over sets of graphs with a fixed number of edges. The main result shows the maximum possible value of $S(G)$ is achieved by three different classes of constructions, depending on the distance between the number of edges and the nearest triangular number. Furthermore we determine the maximum possible value when the set of graphs is restricted to be bipartite, a forest or to be planar given sufficiently many edges. The quantity $S(G)$ corresponds to the difference between two well studied indices, the irregularity of a graph and the sum of the squares of the degrees in a graph. These are known as the first and third Zagreb indices in the area of mathematical chemistry.