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

For a graph G, its energy E(G) is the sum of absolute values of the eigenvalues of its adjacency matrix, the matching number μ(G) is the number of edges in a maximum matching of G, while ∆ is the maximum vertex degree of G. Akbari, Alazemi and Anđelić in Upper Bounds on the Energy of Graphs in Terms of Matching Number. Appl. Anal. Discrete Math. (2021), doi:10.2298/AADM201227016A, proved that E(G) ≤ 2μ(G) √ ∆ holds when G is connected and ∆ ≥ 6, and conjectured that the same inequality is also valid when 2 ≤ ∆ ≤ 5. Here we first computationally enumerate small counterexamples for this conjecture and then provide two infinite families of counterexamples.