三圈图的无符号拉普拉斯谱半径

doi:10.15960/j.cnki.issn.1007-6093.2019.01.009

Abstract

Abstract:

Suppose that the vertex set of a graph G is V(G)={v₁,v₂,…,v_n}. Then we denote by d_{v_i}(G) the degree of v_i in G. Let A(G) be the adjacent matrix of G and D(G) be the n×n diagonal matrix with its (i,i)-entry equal to d_{v_i}(G). Then Q(G)=D(G)+A(G) is the signless Laplacian matrix of G. The signless Laplacian spectral radius of G is the largest eigenvalue of Q(G). In this paper we determine the extremal graph with maximum signless Laplacian spectral radius among all tricyclic graphs of order n.

Key words: signless Laplacian spectral radius, tricyclic graph

CLC Number:

O157.5

CHEN Yuanyuan, WANG Guoping. On the signless Laplacian spectral radius of tricyclic graphs[J]. Operations Research Transactions, 2019, 23(1): 81-89.

References

[1] Zhang H, Zhang X. The Laplacian spectral radius of some bipartite graphs[J]. Linear Algebra and its Applications, 2008, 428:1610-1619.
[2] Abreu N, Cardoso D. On the Laplacian and signless Laplacian spectrum of graph with k pairwise co-neighbor vertices[J]. Linear Algebra and its Applications, 2012, 437:2308-2316.
[3] Ning W, Li H. On the signless Laplacians spectral radius of irregular graphs[J]. Linear Algebra and its Applications, 2013, 438:2280-2288.
[4] Cai G, Fan Y. The signless Laplacians spectral radius of graphs with chromatic number[J]. Mathematica Applicata, 2009, 22:161-167., 2009, 22(1):291-298.
[5] Li J, Shiu W, Chan W. On the Laplacians spectral radii of bipartite graphs[J]. Linear Algebra and its Applications, 2011, 435:2183-2192.
[6] Liu M. The (signless) Laplacian spectral radii of c-cyclic graphs with n vertices and k pendant vertices[J]. Electronic Journal of Linear Algebra, 2012, 23:945-952.
[7] Ye M, Fan Y. Maximizing signless Laplacian or adjacency spectral radius of graphs subject to fixed connectivity[J]. Linear Algebra and its Applications, 2010, 433:1180-1186.
[8] Wang J, Huang Q. Maximizing the signless Laplacians spectral radius of graphs with given diameter or cut vertex[J]. Linear and Multiliner Algebra, 2011, 1-12.
[9] Li K, Wang L. The signless Laplacians spectral radius of tricyclic graphs and trees with k pendant vertices[J]. Linear Algebra and its Applications, 2011, 4354:811-822.
[10] Guo S, Wang Y. The Laplacians spectral radius of tricyclic graphs with n vertices and k pendant vertices[J]. Linear Algebra and its Applications, 2009, 431.
[11] Geng X, Li X. On the index of tricyclic graphs with perfect matchings[J]. Linear Algebra and its Applications, 2009, 431:2304-2316.
[12] Cvetkovi? D, Simic S K. Towards a spectral theory of graphs based on signless Laplacian Ⅱ[J]. Linear Algebra and its Applications, 2010, 432:2257-2272.
[13] Li S, Li X. On tricyclic graphs of a given diameter with minimal energy[J]. Linear Algebra and its Applications, 2009, 430:370-385.
[14] Liu M, Liu B. The signless Laplacian spread[J]. Linear Algebra and its Applications, 2010, 432:505-514.
[15] Liu H, Lu M. On the spectral radius of graphs with cut edges[J]. Linear Algebra and its Applications, 2004, 389:139-145.
[16] Li J, Zhang X. A new upper bound for eigenvalues of the Laplacian matrix of a graph[J]. Linear Algebra and its Applications, 1997, 265:93-100.
[17] Wang J, Huang Q. Some results on the signless Laplacians of graphs[J]. Applied Mathematics Letters, 2010, 23:1045-1049.
[18] Zhang M, Li S. On the signless Laplacian spectra of k-trees[J]. Linear Algebra and its Applications, 2015, 467:136-148.
[19] Hong Y, Zhang X. Sharp upper and lower bounds for largest eigenvalue of the Laplacian matrices of trees[J]. Discrete Mathematics, 2005, 296:187-197.
[20] Cvetkovi? D, Rowlinson P, Simic S K. Signless Laplacian of finite graphs[J]. Linear Algebra and its Applications, 2007, 423:155-171.
[21] Geng X, Li S. The spectral radius of tricyclic graphs with n vertices and k pendant vertices[J]. Linear Algebra and its Applications, 2008, 428:2639-2653.

On the signless Laplacian spectral radius of tricyclic graphs

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[5]	CHEN Lin, HUANG Qiongxiang. Tricyclic graphs with exactly two Q-main eigenvalues [J]. Operations Research Transactions, 2014, 18(3): 13-32.
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