Note on the energy of regular graphs

For a simple graph $G$, the energy $\mathcal{E}(G)$ is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix $A(G)$. Let $n, m$, respectively, be the number of vertices and edges of $G$. One well-known inequality is that $\mathcal{E}(G)\leq \lambda_1+\sqrt{(n-1)(2m-\lambda_1)}$, where $\lambda_1$ is the spectral radius. If $G$ is $k$-regular, we have $\mathcal{E}(G)\leq k+\sqrt{k(n-1)(n-k)}$. Denote $\mathcal{E}_0=k+\sqrt{k(n-1)(n-k)}$. Balakrishnan [{\it Linear Algebra Appl.} {\bf 387} (2004) 287--295] proved that for each $\epsilon>0$, there exist infinitely many $n$ for each of which there exists a $k$-regular graph $G$ of order $n$ with $k< n-1$ and $\frac{\mathcal{E}(G)}{\mathcal{E}_0}<\epsilon$, and proposed an open problem that, given a positive integer $n\geq 3$, and $\epsilon>0$, does there exist a $k$-regular graph $G$ of order $n$ such that $\frac{\mathcal{E}(G)}{\mathcal{E}_0}>1-\epsilon$. In this paper, we show that for each $\epsilon>0$, there exist infinitely many such $n$ that $\frac{\mathcal{E}(G)}{\mathcal{E}_0}>1-\epsilon$. Moreover, we construct another class of simpler graphs which also supports the first assertion that $\frac{\mathcal{E}(G)}{\mathcal{E}_0}<\epsilon$.