The minimal distance signless Laplacian spectral radii of connected graphs with given clique number
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摘要: 在本文中, 我们刻画了给定团数的连通图中取得最小距离无符号拉普拉斯谱半径的极图.
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关键词:
- 谱半径
Abstract: In this paper, we characterize the extremal graph with the minimal distance signless Laplacian spectral radius among all connected graphs with given clique number.-
Keywords:
- spectral radius
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[1] M. Aouchiche, P. Hansen, Two Laplacians for the distance matrix of a graph,
Linear Algebra Appl. 439 (2013) 21-33.
[2] J.A. Bondy, U.S.R. Murty, Graph Theory With Applications, Macmillan, Lon-
don, 1976.
[3] J.A. Bondy, U.S.R. Murty, Graph Theory, Springer, New York, 2008.
[4] Abraham Berman, Robert J. Plemmons, Nonnegative Matrices in the Mathe-
matical Sciences, New York: Academic Press, 1979.
[5] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University, England,
1986.
[6] H. Minc, Nonnegative Matrices, John Wiley and Sons Inc., New York, 1988.
[7] J. Bang{Jensen, G. Gutin, Digraphs Theory, Algorithms and Applications,
Springer, 2001.
[8] R.L. Graham, L. Lovasz, Distance matrix polynomials of trees, Adv.Math 29
(1978) 60-88.
[9] G. Indulal, Sharp bounds on the distance spectral radius and the distance energy
of graphs, Linear Algebra Appl. 430 (2009) 106-133.
[10] D. Stevanovic, A. Ilic, Distance spectral radius of trees with xed maximum
degree, Electron. J. Linear Algebra. 20 (2010) 168-179.
[11] Z.Z. Liu, On spectral radius of the distance matrix, Appl. Anal. Discrete Math.
4 (2010) 269-277.
[12] X.L. Zhang, C. Godsil, Connectivity and minimal distance spectral radius of
graphs, Linear and Multilinear Algebra 59 (2011) 745-754.
[13] G.L. Yu, Y.R. Wu, Y.J. Zhang, J.L. Shu, Some graft transformations and its
application on a distance spectrum, Discrete Math. 311 (2011) 2117-2123.
[14] S.S. Bose, M. Nath, S. Paul, Distance spectral radius of graphs with r pendent
vertices, Linear Algebra Appl. 435 (2011) 2828-2836.
[15] G.L. Yu, H.C. Jia, H.L. Zhang, J.L. Shu, Some graft transformations and its
applications on the distance spectral radius of a graph, Applied Mathematics
Letters 25 (2012) 315-319.
[16] X.L. Zhang, On the distance spectral radius of some graphs, Linear Algebra
Appl. 437 (2012) 1930-1941.
[17] M. Nath, S. Paul, On the distance spectral radius of bipartite graphs, Linear
Algebra Appl. 436 (2012) 1285-1296.
[18] M.Q. Zhai, G.L. Yu, J.L. Shu, Clique number and distance spectral radii of
graphs, Ars Combin. 104 (2012) 385-392.
[19] Y.N. Wang, B. Zhou, On distance spectral radius of graphs, Linear Algebra
Appl. 438 (2013) 3490-3503.
[20] W.J. Ning, L.Q. Ouyang, M. Lu, Distance spectral radius of trees with xed
number of pendent vertices, Linear Algebra Appl. 439 (2013) 2240-2249.
[21] Y.Y. Chen, H.Q. Lin, J.L. Shu, Sharp upper bounds on the distance spectral
radius of a graph, Linear Algebra Appli. 439 (2013) 2659-2666.
[22] H.Q. Lin, J.L. Shu, Sharp bounds on distance spectral radius of graphs, Linear
and Multilinear Algebra 61 (2013) 442-447.
[23] R.D. Xing, B. Zhou, J.P. Li, On the distance signless Laplacian spectral radius
of graphs, Linear and Multilinear Algebra 62 (2014) 1377-1387.
[24] R.D. Xing, B. Zhou, On the distance and distance signless Laplacian spectral
radii of bicyclic graphs, Linear Algebra Appl. 439 (2013) 3955-3963.
[25] F.L. Tian, D. Wong, J.L. Rou, Proof for four conjectures about the distance
Laplacian and distance signless Laplacian eigenvalues of a graph, Linear Algebra
Appl. 471 (2015) 10-20.
[26] K.C. Das, Proof of conjectures on the distance signless Laplacian eigenvalues
of graphs, Linear Algebra Appl. 467 (2015) 100-115.
[27] D. Stevanovic, P. Hansen, The minimum spectral radius of graphs with a given
clique number, Electron.J.Linear Algebra 17 (2008) 110-117.
[28] H.Q. Lin, J.L. Shu, Y.R. Wu, G.L. Yu, Spectral radius of strongly connected
digraphs, Discrete Math. 312 (2012) 3663-3669.
[29] J.M. Guo, J.X. Li, W.C. Shiu, The smallest Laplacian spectral radius of graphs
with a given clique number, Linear Algebra Appl. 437 (2012) 1109-1122.
[30] B. He, Y.L. Jin, X.D. Zhang, Sharp bounds for the signless Laplacian spectral
radius in terms of clique number, Linear Algebra Appl. 438 (2013) 3851-3861.
[31] J.M. Zhang, T.Z. Huang, J.M. Guo, The smallest signless Laplacian spectral
radius of graphs with a given clique number, Linear Algebra Appl. 439 (2013)
2562-2576.
[32] W.X. Hong, L.H. You, Spectral radius and signless Laplacian spectral radius
of strongly connected digraphs, Linear Algebra Appl. 457 (2014) 93-113.
[33] B. Bollobas, Extremal graph theory, Academic Press, London, New York, San
Francisco, 1978.
[34] M. Kang, O. Pikhurko, Maximum Kr+1-free graphs which are not r-partite,
Mat. Stud. 24 (2005) 12-20.[1] M. Aouchiche, P. Hansen, Two Laplacians for the distance matrix of a graph,
Linear Algebra Appl. 439 (2013) 21-33.
[2] J.A. Bondy, U.S.R. Murty, Graph Theory With Applications, Macmillan, Lon-
don, 1976.
[3] J.A. Bondy, U.S.R. Murty, Graph Theory, Springer, New York, 2008.
[4] Abraham Berman, Robert J. Plemmons, Nonnegative Matrices in the Mathe-
matical Sciences, New York: Academic Press, 1979.
[5] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University, England,
1986.
[6] H. Minc, Nonnegative Matrices, John Wiley and Sons Inc., New York, 1988.
[7] J. Bang{Jensen, G. Gutin, Digraphs Theory, Algorithms and Applications,
Springer, 2001.
[8] R.L. Graham, L. Lovasz, Distance matrix polynomials of trees, Adv.Math 29
(1978) 60-88.
[9] G. Indulal, Sharp bounds on the distance spectral radius and the distance energy
of graphs, Linear Algebra Appl. 430 (2009) 106-133.
[10] D. Stevanovic, A. Ilic, Distance spectral radius of trees with xed maximum
degree, Electron. J. Linear Algebra. 20 (2010) 168-179.
[11] Z.Z. Liu, On spectral radius of the distance matrix, Appl. Anal. Discrete Math.
4 (2010) 269-277.
[12] X.L. Zhang, C. Godsil, Connectivity and minimal distance spectral radius of
graphs, Linear and Multilinear Algebra 59 (2011) 745-754.
[13] G.L. Yu, Y.R. Wu, Y.J. Zhang, J.L. Shu, Some graft transformations and its
application on a distance spectrum, Discrete Math. 311 (2011) 2117-2123.
[14] S.S. Bose, M. Nath, S. Paul, Distance spectral radius of graphs with r pendent
vertices, Linear Algebra Appl. 435 (2011) 2828-2836.
[15] G.L. Yu, H.C. Jia, H.L. Zhang, J.L. Shu, Some graft transformations and its
applications on the distance spectral radius of a graph, Applied Mathematics
Letters 25 (2012) 315-319.
[16] X.L. Zhang, On the distance spectral radius of some graphs, Linear Algebra
Appl. 437 (2012) 1930-1941.
[17] M. Nath, S. Paul, On the distance spectral radius of bipartite graphs, Linear
Algebra Appl. 436 (2012) 1285-1296.
[18] M.Q. Zhai, G.L. Yu, J.L. Shu, Clique number and distance spectral radii of
graphs, Ars Combin. 104 (2012) 385-392.
[19] Y.N. Wang, B. Zhou, On distance spectral radius of graphs, Linear Algebra
Appl. 438 (2013) 3490-3503.
[20] W.J. Ning, L.Q. Ouyang, M. Lu, Distance spectral radius of trees with xed
number of pendent vertices, Linear Algebra Appl. 439 (2013) 2240-2249.
[21] Y.Y. Chen, H.Q. Lin, J.L. Shu, Sharp upper bounds on the distance spectral
radius of a graph, Linear Algebra Appli. 439 (2013) 2659-2666.
[22] H.Q. Lin, J.L. Shu, Sharp bounds on distance spectral radius of graphs, Linear
and Multilinear Algebra 61 (2013) 442-447.
[23] R.D. Xing, B. Zhou, J.P. Li, On the distance signless Laplacian spectral radius
of graphs, Linear and Multilinear Algebra 62 (2014) 1377-1387.
[24] R.D. Xing, B. Zhou, On the distance and distance signless Laplacian spectral
radii of bicyclic graphs, Linear Algebra Appl. 439 (2013) 3955-3963.
[25] F.L. Tian, D. Wong, J.L. Rou, Proof for four conjectures about the distance
Laplacian and distance signless Laplacian eigenvalues of a graph, Linear Algebra
Appl. 471 (2015) 10-20.
[26] K.C. Das, Proof of conjectures on the distance signless Laplacian eigenvalues
of graphs, Linear Algebra Appl. 467 (2015) 100-115.
[27] D. Stevanovic, P. Hansen, The minimum spectral radius of graphs with a given
clique number, Electron.J.Linear Algebra 17 (2008) 110-117.
[28] H.Q. Lin, J.L. Shu, Y.R. Wu, G.L. Yu, Spectral radius of strongly connected
digraphs, Discrete Math. 312 (2012) 3663-3669.
[29] J.M. Guo, J.X. Li, W.C. Shiu, The smallest Laplacian spectral radius of graphs
with a given clique number, Linear Algebra Appl. 437 (2012) 1109-1122.
[30] B. He, Y.L. Jin, X.D. Zhang, Sharp bounds for the signless Laplacian spectral
radius in terms of clique number, Linear Algebra Appl. 438 (2013) 3851-3861.
[31] J.M. Zhang, T.Z. Huang, J.M. Guo, The smallest signless Laplacian spectral
radius of graphs with a given clique number, Linear Algebra Appl. 439 (2013)
2562-2576.
[32] W.X. Hong, L.H. You, Spectral radius and signless Laplacian spectral radius
of strongly connected digraphs, Linear Algebra Appl. 457 (2014) 93-113.
[33] B. Bollobas, Extremal graph theory, Academic Press, London, New York, San
Francisco, 1978.
[34] M. Kang, O. Pikhurko, Maximum Kr+1-free graphs which are not r-partite,
Mat. Stud. 24 (2005) 12-20.
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