Some Relations between Rank, Chromatic Number and Energy of Graphs

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 $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(G)$ be the rank of the adjacency matrix of $G$. In this paper we characterize all graphs with $E(G)=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(G)$. Among other results we show that apart from a few families of graphs, $E(G)\geq 2\max(\chi(G), n-\chi(\bar{G}))$,where$n$isthenumberofverticesof$G$,$\bar{G}$and$\chi(G)$arethecomplementandthechromaticnumberof$G$, respectively. Moreover some new lower bounds for $E(G)$ in terms of $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(G)$ are given.