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Chromatic number and signless Laplacian spectral radius of graphs | ||
| Transactions on Combinatorics | ||
| مقاله 6، دوره 11، شماره 4، اسفند 2022، صفحه 327-334 اصل مقاله (444.37 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2021.129720.1876 | ||
| نویسنده | ||
| Mohammad Reza Oboudi* | ||
| Department of Mathematics, College of Sciences, Shiraz University, Shiraz, 71457-44776, Iran | ||
| چکیده | ||
| 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. | ||
| کلیدواژه‌ها | ||
| Chromatic number؛ Majorization؛ Signless Laplacian matrix؛ Signless Laplacian spectral radius | ||
| مراجع | ||
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[7] M. R. Oboudi, A relation between the signless Laplacian spectral radius of complete multipartite graphs and majorization, Linear Algebra Appl., 565 (2019) 225–238. [8] M. R. Oboudi, Distance spectral radius of complete multipartite graphs and majorization, Linear Algebra Appl., 583 (2019) 134–145. [9] M. R. Oboudi, Majorization and the spectral radius of starlike trees, J. Comb. Optim., 36 (2018) 121–129.
[10] M. R. Oboudi, On the difference between the spectral radius and maximum degree of graphs, Algebra Discrete Math., 24 (2017) 302–307. [11] M. R. Oboudi, On the largest real root of independence polynomials of graphs, an ordering on graphs, and starlike trees, arXiv:1303.3222. [12] M. R. Oboudi, On the third largest eigenvalue of graphs, Linear Algebra Appl., 503 (2016) 164–179.
[13] H. S. Wilf, Spectral bounds for the clique and independence numbers of graphs, J. Combin. Theory Ser. B, 40 (1986) 113–117. [14] H. S. Wilf, The eigenvalues of a graph and its chromatic number, J. London Math. Soc., 42 (1967) 330–332.
[15] G. Yu, Y. Wu and J. Shu, Signless Laplacian spectral radii of graphs with given chromatic number, Linear Algebra Appl., 435 (2011) 1813–1822. | ||
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