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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 thesum 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 thispaper we propose a stronger conjecture as for $n \geq 5$, $\mathcal{E}(G)\geqn-1+ d$, where $d$ is the average degree of $G$. Here, we show that conjectureholds for bipartite graphs, planar graphs and for the graphs with $d \leqn-2\ln n -3$

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