A lower bound of the energy of non-singular graphs in terms of average degree

Let $G$ be a graph of order $n$ with adjacency matrix $A(G)$. The $\mathit{\text{energy}}$ of graph $G$, denoted by $\mathcal{E}(G)$, is defined as the sum of absolute value of eigenvalues of $A(G)$. It was conjectured that if $A(G)$ is non-singular, then $\mathcal{E}(G)\ge \mathrm{\Delta }(G)+\delta (G)$. In this paper we propose a stronger conjecture as for $n\ge 5$, $\mathcal{E}(G)\ge n-1+d$, where $d$ is the average degree of $G$. Here, we show that conjecture holds for bipartite graphs, planar graphs and for the graphs with $d\le n-2\ln n-3$