Rank-Average Degree Bound for Graph Energy

We prove that the energy $\mathcal{E}(G)$ of any simple graph $G$ of order $n\ge 5$ satisfies $$\mathcal{E}\ge r(G)+\bar{d}(G)-1,$$ where $r(G)$ and $\bar{d}(G)$ denote, respectively, the rank of the adjacency matrix and the average degree of $G$. We also characterize all extremal graphs. As consequences, our result settles five previously conjectured lower bounds for the energy of nonsingular graphs in their stated ranges, namely $$\begin{array}{c}{\displaystyle \mathcal{E}(G)\ge n-1+\bar{d}(G),\mathcal{E}(G)\ge \mathrm{\Delta }(G)+\delta (G),\mathcal{E}(G)\ge 2\sqrt{\bar{d}(G)(n-1)},\mathcal{E}(G)\ge \frac{M_1(G)}{m},\mathcal{E}(G)\ge \frac{M_1(G)}{2m}+\frac{2m}{n},}\end{array}$$ where $m$ is the number of edges, $\mathrm{\Delta }(G)$ and $\delta (G)$ are the maximum and minimum degrees, and the first Zagreb index $M_1(G)$ is the sum of degree squares.