A proof of the conjecture on hypoenergetic graphs with maximum degree $\mathrm{\Delta }\le 3$

The energy $E(G)$ of a graph $G$ is defined as the sum of the absolute values of its eigenvalues. A graph $G$ of order $n$ is said to be hypoenergetic if $E(G)<n$. Majstorovi'{c} et al. conjectured that complete bipartite graph $K_{2,3}$ is the only hypoenergetic connected quadrangle-containing graph with maximum degree $\mathrm{\Delta }\le 3$. This paper is devoted to giving a confirmative proof to the conjecture.