  Let $G$ be a graph of order $n$ with adjacency matrix $A(G)$. The
\textit{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 $\mathcal{E}(G)\geq\Delta(G)+\delta(G)$. In this
paper we propose a stronger conjecture as for $n \geq 5$, $\mathcal{E}(G)\geq
n-1+ d$, where $d$ is the average degree of $G$. Here, we show that conjecture
holds for bipartite graphs, planar graphs and for the graphs with $d \leq
n-2\ln n -3$
