All Connected Graphs with Maximum Degree at Most 3 whose Energies are Equal to the Number of Vertices

The energy $E(G)$ of a graph $G$ is defined as the sum of the absolute values of its eigenvalues. Let $S_2$ be the star of order 2 (or $K_2$) and $Q$ be the graph obtained from $S_2$ by attaching two pendent edges to each of the end vertices of $S_2$. Majstorovi'c et al. conjectured that $S_2$, $Q$ and the complete bipartite graphs $K_{2,2}$ and $K_{3,3}$ are the only 4 connected graphs with maximum degree $\mathrm{\Delta }\le 3$ whose energies are equal to the number of vertices. This paper is devoted to giving a confirmative proof to the conjecture.