Abstract
Let \(\fancyscript {P}_{n}^{r}\) denote the set of all connected graphs of order n with r pendant vertices. In this paper, we determine the unique graph with maximal signless Laplacian spectral radius among all graphs in \(\fancyscript {P}_{n}^{r}\). In addition, we determine the unique graph with minimal signless Laplacian spectral radius among all graphs in \(\fancyscript {P}_{n}^{r}\) for each r ∈ {0,1,2,3, n − 2, n − 1}.
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References
Anderson, W.N., Morely, T.D.: Eigenvalues of the Laplacian of a graph. Linear Multilinear Algebra 18, 141–145 (1985)
Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. American Elsevier Publishing Co., New York (1976)
Cvertkoć, D.: Signless Laplacians and line graphs. Bull. Acad. Serbe Sci. Arts. Clin. Sci. Math. Nat. Sci. Math. 131(30), 85–92 (2005)
Cvetković, D., Rowlinson, P., Simić, S.: Signless Laplacians of finite graphs. Linear Algebra Appl. 423, 155–171 (2007)
van Dam, E.R., Haemers, W.: Which graphs are determined by their spectrum? Linear Algebra Appl. 373, 241–272 (2003)
Desai, M., Rao, V.: A characterization of the smallest eigenvalue of a graph. J. Graph Theory 18, 181–194 (1994)
Fan, Y.-Z., Tam, B.-S., Zhou, J.: Maximizing spectral radius of unoriented Laplacian matrix over bicyclic graphs of a given order. Linear Multilinear Algebra 56(4), 381–397 (2008)
Feng, L.-H., Li, Q., Zhang, X.-D.: Minimizing the Laplacian spectral radius of trees with given matching number. Linear Multilinear Algebra 55(2), 199–207 (2007)
Fiedler, M.: A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory. Czechoslovak Math. J. 25, 607–618 (1975)
Grossman, J.W., Kulkarni, D.M., Schochetman, I.: Algebraic graph theory without orientation. Linear Algebra Appl. 212/213, 289–307 (1994)
Haemers, W., Spence, E.: Enumeration of cospectral graphs. Eur. J. Comb. 25, 199–211 (2004)
Hong, Y., Zhang, X.-D.: Sharp upper and lower bounds for largest eigenvalue of the Laplacian matrices of trees. Discrete Math. 296, 187–197 (2005)
Li, Q., Feng, K.: On the largest eigenvalue of a graph (in Chinese). Acta Math. Appl. Sin. 2, 167–175 (1979)
Merris, R.: Laplacian matrices of graphs: a survey. Linear Algebra Appl. 197/198, 143–176 (1994)
Tam, B.-S., Fan, Y.-Z., Zhou, J.: Unoriented Laplacian maximizing graphs are degree maximal. Linear Algebra Appl. 429(4), 735–758 (2008)
Wu, B., Xiao, E., Hong, Y.: The spectral radius of trees on k pendant vertices. Linear Algebra Appl. 395, 343–349 (2005)
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Supported by National Natural Science Foundation of China (10601001), Anhui Provincial Natural Science Foundation (070412065), NSF of Department of Education of Anhui Province (2005kj005zd), Project of Anhui University on Leading Researchers Construction, Foundation of Innovation Team on Basic Mathematics of Anhui University.
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Fan, YZ., Yang, D. The Signless Laplacian Spectral Radius of Graphs with Given Number of Pendant Vertices. Graphs and Combinatorics 25, 291–298 (2009). https://doi.org/10.1007/s00373-009-0840-1
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DOI: https://doi.org/10.1007/s00373-009-0840-1