On the Zagreb Indices Equality

TL;DR

This paper investigates the conditions under which certain graphs satisfy the Zagreb indices equality, extending previous results to graphs with higher maximum degree and considering degree constraints.

Contribution

It generalizes prior findings by proving the existence of infinitely many graphs with maximum degree at least 5 satisfying the Zagreb indices equality, beyond regular and biregular graphs.

Findings

Existence of infinitely many graphs with maximum degree ≥ 5 satisfying the Zagreb indices equality.

Characterization of graphs with degrees in a prescribed interval that satisfy the Zagreb indices equality.

Extension of previous results from degree 4 to higher maximum degrees.

Abstract

For a simple graph $G$ with $n$ vertices and $m$ edges, the first Zagreb index and the second Zagreb index are defined as $M_1(G)=\sum_{v\in V}d(v)^2 $ and $M_2(G)=\sum_{uv\in E}d(u)d(v)$. In \cite{VGFAD}, it was shown that if a connected graph $G$ has maximal degree 4, then $G$ satisfies $M_1(G)/n = M_2(G)/m$ (also known as the Zagreb indices equality) if and only if $G$ is regular or biregular of class 1 (a biregular graph whose no two vertices of same degree are adjacent). There, it was also shown that there exist infinitely many connected graphs of maximal degree $\Delta= 5$ that are neither regular nor biregular of class 1 which satisfy the Zagreb indices equality. Here, we generalize that result by showing that there exist infinitely many connected graphs of maximal degree $\Delta \geq 5$ that are neither regular nor biregular graphs of class 1 which satisfy the Zagreb indices equality. We also consider when the above equality holds when the degrees of vertices of a given graph are in a prescribed interval of integers.