Akbari–Alazemi–Anđelić's energy–matching number conjecture

Let $G$not be isomorphic to $C_3$, $C_5$, or $C_7$, where $G$ is a connected graph, and let $\Delta(G)$ denote its maximum vertex degree. Let $\mathcal{E}(G)$ be the graph energy and $\nu(G)$ the matching number.

Akbari–Alazemi–Anđelić's conjecture. For any connected graph $G \not\cong C_3, C_5, C_7$ with $\Delta(G) \in \{ 2, 3, 4, 5 \}$, we have

$$\mathcal{E}(G) \le 2 \nu(G) \sqrt{\Delta(G)}.$$

This conjecture asks whether the maximum-degree hypothesis in the known theorem can be relaxed. It was refuted using Wagner's original approach through infinitely many counterexamples.