On a Conjecture about Comparing the First and Second Zagreb Indices of Graphs

Abstract

Let $G$ be a graph with order $n(G)$, size $m(G)$, first Zagreb index $M_1(G)$, and second Zagreb index $M_2(G)$. More than twenty years ago, it was conjectured that $\frac{M_1(G)}{n(G)} \leq \frac{M_2(G)}{m(G)}$. Later, Hansen and Vuki\v{c}evi\'c demonstrated that this conjecture does not hold for general graphs but is valid for chemical graphs. In this paper, as an extension of the study of chemical graphs, we investigate graphs in which the difference between the minimum and maximum degrees is at most $3$. We prove that any graph in this class that serves as a counterexample to the stated conjecture must have a minimum degree of $2$ and a maximum degree of $5$. Furthermore, we present infinitely many connected graphs that serve as counterexamples to this conjecture, all of which have a minimum degree of 2, a maximum degree of 5, and an order of at least 218.