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The energy of a graph $G$, denoted by $E(G)$, is defined as the
sum of the absolute values of all eigenvalues of $G$. Let $G$ be a
graph of order $n$ and rank $(G)$ be the rank of the adjacency
matrix of $G$. In this paper we characterize all graphs with $E(G)
=$ rank$(G)$. Among other results we show that apart from a few
families of graphs, $E(G) \geq 2max(\mathcal{X}(G),n -
\mathcal{X}(\bar{G})$, where $n$ is the number of vertices of $G$,
$\bar{G}$ and $\mathcal{X}(G)$ are the complement and the
chromatic number of $G$, respectively. Moreover some new lower
bounds for $E(G)$ in terms of rank $(G)$ are given.
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