Finding counterexamples for a conjecture of Akbari, Alazemi and Andjeli\'c

TL;DR

This paper investigates a conjecture relating graph energy and matching number, providing computational enumeration of small counterexamples and introducing two infinite families of counterexamples to disprove the conjecture.

Contribution

The paper offers the first known counterexamples to the conjecture for 2 ≤ Δ ≤ 5, including infinite families, challenging previous assumptions.

Findings

Counterexamples for 2 ≤ Δ ≤ 5 are identified.

Two infinite families of graphs serve as counterexamples.

The conjecture does not hold universally for all connected graphs with Δ ≥ 2.

Abstract

For a graph $G$, its energy $\mathcal{E}(G)$ is the sum of absolute values of the eigenvalues of its adjacency matrix, the matching number $\mu(G)$ is the number of edges in a maximum matching of $G$, while $\Delta$ is the maximum vertex degree of $G$. Akbari, Alazemi and An{\dj}eli\'c in [Appl. Anal. Discrete Math. 15 (2021), 444--459] proved that $\mathcal{E}(G) \leq 2\mu(G)$ when $G$ is connected and $\Delta\geq6$, and conjectured that the same inequality is also valid when $2\leq\Delta\leq5$. Here we first computationally enumerate small counterexamples for this conjecture and then provide two infinite families of counterexamples.