Chromatic number and signless Laplacian spectral radius of graphs
For any simple graph $G$, the signless Laplacian matrix of $G$ is defined as $D(G)+A(G)$, where $D(G)$ and $A(G)$ are the diagonal matrix of vertex degrees and the adjacency matrix of $G$, respectively. %Let $chi(G)$ be the chromatic number of $G$ Let $q(G)$ be the signless Laplacian spectral radius of $G$ (the largest eigenvalue of the signless Laplacian matrix of $G$). In this paper we find some relations between the chromatic number and the signless Laplacian spectral radius of graphs. In particular, we characterize all graphs $G$ of order $n$ with odd chromatic number $chi$ such that $q(G)=2nBig(1-frac{1}{chi}Big)$. Finally we show that if $G$ is a graph of order $n$ and with chromatic number $chi$, then under certain conditions, $q(G)<2nBig(1-frac{1}{chi}Big)-frac{2}{n}$. This result improves some previous similar results.
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