The signless Laplacian spread. (English) Zbl 1206.05064

The signless Laplacian spread of \(G\) is defined as \(SQ(G) = \mu_1 (G) - \mu_n (G)\), where \(\mu_1 (G)\) and \(\mu_n (G)\) are the maximum and minimum eigenvalues of the signless Laplacian matrix of \(G\), respectively. This paper presents some upper and lower bounds for \(SQ(G)\). Moreover, the unique unicyclic graph with maximum signless Laplacian spread among the class of connected unicyclic graphs of order \(n\) is determined.


05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
15A42 Inequalities involving eigenvalues and eigenvectors
15B36 Matrices of integers
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[1] Cardoso, D.M.; Cvetković, D.; Rowlinson, P.; Simić, S.K., A sharp lower bound for the least eigenvalue of the signless Laplacian of a non-bipartite graph, Linear algebra appl., 429, 2770-2780, (2008) · Zbl 1148.05046
[2] Cvetković, D.M.; Doob, M.; Sachs, H., Spectra of graphs-theory and applications, (1980), V.E.B. Deutscher Verlag der Wissenschaften Berlin · Zbl 0458.05042
[3] Cvetković, D.; Rowlinson, P.; Simić, S.K., Signless Laplacians of finite graphs, Linear algebra appl., 423, 155-171, (2007) · Zbl 1113.05061
[4] Dai, H., Matrix theory, (2001), Science Press Beijing, (in Chinese)
[5] Das, K.C., The Laplacian spectrum of a graph, Comput. math. appl., 48, 715-724, (2004) · Zbl 1058.05048
[6] Fan, Y.Z., Largest eigenvalue of a unicyclic mixed graph, Appl. math. J. Chinese univ. ser. B., 19, 2, 140-148, (2004) · Zbl 1059.05072
[7] Fan, Y.Z.; Tam, B.S.; Zhou, J., Maximizing spectral radius of unoriented Laplacian matrix over bicyclic graphs of a given order, Linear and multilinear algebra, 56, 4, 381-397, (2008) · Zbl 1146.05032
[8] Fan, Y.Z.; Wang, Y.; Gao, Y.B., Minimizing the least eigenvalues of unicyclic graphs with application to spectral spread, Linear algebra appl., 429, 2-3, 577-588, (2008) · Zbl 1143.05053
[9] Fan, Y.Z.; Xu, J.; Wang, Y.; Liang, D., The Laplacian spread of a tree, Disc. math. theor. comput. sci., 10, 1, 79-86, (2008) · Zbl 1153.05323
[10] Gregory, D.A.; Hershkowitz, D.; Kirkland, S.J., The spread of the spectrum of a graph, Linear algebra appl., 332-334, 23-35, (2001) · Zbl 0978.05049
[11] Grone, R., Eigenvalues and the degree sequences of graphs, Linear and multilinear algebra, 39, 133-136, (1995) · Zbl 0831.05047
[12] Haemers, W.H., Interlacing eigenvalues and graphs, Linear algebra appl., 227-228, 593-616, (1995) · Zbl 0831.05044
[13] Li, J.X.; Shiu, W.C.; Chan, W.H., Some results on the Laplacian eigenvalues of unicyclic graphs, Linear algebra appl., 430, 2080-2093, (2009) · Zbl 1225.05169
[14] M.H. Liu, On the Laplacian spread of trees and unicyclic graphs, Comput. Math. Appl., submitted for publication. · Zbl 1265.05397
[15] M.H. Liu, B.L. Liu, On the signless Laplacian spectral radii of bicyclic and tricyclic graphs, Disc. Math., submitted for publication. · Zbl 1349.05213
[16] Liu, M.H.; Liu, B.L., New sharp upper bounds for the first Zagreb index, MATCH commun. math. comput. chem., 62, 689-698, (2009) · Zbl 1224.05117
[17] Liu, B.L.; Liu, M.H., On the spread of the spectrum of a graph, Disc. math., 309, 2727-2732, (2009) · Zbl 1194.05091
[18] M.H. Liu, X.Z. Tan, B.L. Liu, On the ordering of the signless Laplacian spectral radii of unicyclic graphs, Appl. Math. J. Chinese Univ. Ser. B., in press. · Zbl 1228.05208
[19] Marshall, A.W.; Olkin, I., Inequalities: theory of majorization and its applications, (1979), Academic New York · Zbl 0437.26007
[20] Merris, R., Laplacian matrices of graphs: a survey, Linear algebra appl., 197-198, 143-176, (1994) · Zbl 0802.05053
[21] Pan, Y.L., Sharp upper bounds for the Laplacian graph eigenvalues, Linear algebra appl., 355, 287-295, (2002) · Zbl 1015.05055
[22] Petrović, M., On graphs whose spectral spread does not exceed 4, Publ. inst. math., 34, 48, 169-174, (1983) · Zbl 0554.05046
[23] Petrović, M.; Borovićanin, B.; Aleksić, T., Bicyclic graphs for which the least eigenvalue is minimum, Linear algebra appl., 430, 4, 1328-1335, (2009) · Zbl 1194.05093
[24] Tam, B.S.; Fan, Y.Z.; Zhou, J., Unoriented Laplacian maximizing graphs are degree maximal, Linear algebra appl., 429, 735-758, (2008) · Zbl 1149.05034
[25] F.Y. Wei, M.H. Liu, More results on the ordering of the signless Laplacian spectral radii of unicyclic graphs, Disc. Math. Theoet. Comput. Sci., submitted for publication. · Zbl 1228.05208
[26] F.Y. Wei, M.H. Liu, On the signless Laplacian spectral radii of bicyclic graphs, Electron. Linear Algebra, submitted for publication.
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