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arXiv · 2111.15303

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

Đorđe Stevanović·Ivan Damnjanović·Dragan Stevanović
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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.

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Đorđe Stevanović, Ivan Damnjanović, Dragan Stevanović. 2021-11-30. Finding counterexamples for a conjecture of Akbari, Alazemi and Andjeli\'c. https://arxiv.org/abs/2111.15303

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