%0 Journal Article
%T Chromatic number and signless Laplacian spectral radius of graphs
%J Transactions on Combinatorics
%I University of Isfahan
%Z 2251-8657
%A Oboudi, Mohammad Reza
%D 2022
%\ 12/01/2022
%V 11
%N 4
%P 327-334
%! Chromatic number and signless Laplacian spectral radius of graphs
%K chromatic number
%K Majorization
%K Signless Laplacian matrix
%K Signless Laplacian spectral radius
%R 10.22108/toc.2021.129720.1876
%X 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 $q(G)=2n\Big(1-\frac{1}{\chi}\Big)$. 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.
%U https://toc.ui.ac.ir/article_26159_fec709a44b419ec4680f4424fb080a49.pdf