  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 $\Delta \leq 3$ whose energies are equal to the
number of vertices. This paper is devoted to giving a confirmative proof to the
conjecture.
