University of IsfahanTransactions on Combinatorics2251-865711420221201Chromatic number and signless Laplacian spectral radius of graphs3273342615910.22108/toc.2021.129720.1876ENMohammad RezaOboudiDepartment of Mathematics, College of Sciences, Shiraz University, Shiraz, 71457-44776, IranJournal Article20210723For 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 $q(G)=2n\Big(1-\frac{1}{\chi}\Big)$. Finally we show that if $G$ is a graph of order $n$ and with chromatic number $\chi$, then under certain conditions, $q(G)<2n\Big(1-\frac{1}{\chi}\Big)-\frac{2}{n}$. This result improves some previous similar results.https://toc.ui.ac.ir/article_26159_fec709a44b419ec4680f4424fb080a49.pdf