On comparing Zagreb indices

TL;DR

This paper investigates the bounds and validity of a conjecture relating Zagreb indices in graphs, demonstrating conditions under which the conjecture holds or fails, and establishing new bounds for these topological indices.

Contribution

The paper provides new bounds for Zagreb indices, shows the conjecture's limitations based on graph cycles and degrees, and proves the conjecture for subdivision graphs.

Findings

Bounds for $M_1/n$ and $M_2/m$ are the same and achieved only by regular graphs.

Counterexamples exist where the conjecture fails for graphs with multiple cycles.

The conjecture holds for subdivision graphs.

Abstract

Let $G=(V,E)$ be a simple graph with $n = |V|$ vertices and $m = |E|$ edges. The first and second Zagreb indices are among the oldest and the most famous topological indices, defined as $M_1 = \sum_{i \in V} d_i^2$ and $M_2 = \sum_{(i, j) \in E} d_i d_j$, where $d_i$ denote the degree of vertex $i$. Recently proposed conjecture $M_1 / n \leqslant M_2 / m$ has been proven to hold for trees, unicyclic graphs and chemical graphs, while counterexamples were found for both connected and disconnected graphs. Our goal is twofold, both in favor of a conjecture and against it. Firstly, we show that the expressions $M_1/n$ and $M_2/m$ have the same lower and upper bounds, which attain equality for and only for regular graphs. We also establish sharp lower bound for variable first and second Zagreb indices. Secondly, we show that for any fixed number $k\geqslant 2$, there exists a connected graph with $k$ cycles for which $M_1/n>M_2/m$ holds, effectively showing that the conjecture cannot hold unless there exists some kind of limitation on the number of cycles or the maximum vertex degree in a graph. In particular, we show that the conjecture holds for subdivision graphs.