Akbari–Alazemi–Andjelić energy bound conjecture for graphs

Let $G$ be a connected graph with adjacency eigenvalues $\lambda_1\geq\cdots\geq\lambda_N$, energy $\mathcal{E}(G)=\sum_{i=1}^N|\lambda_i|$, matching number $\mu(G)$, and largest vertex degree $\Delta$.

Akbari–Alazemi–Andjelić conjecture. The inequality

$$\mathcal{E}(G)\leq 2\mu(G)\sqrt{\Delta}$$

holds for any connected graph $G\not\cong C_3,C_5,C_7$ with $\Delta\in\{2,3,4,5\}$.

The conjecture extends a theorem proving the bound for connected graphs with $\Delta\geq 6$. It is refuted: the paper constructs two infinite families of counterexamples.