Abstract
Let K be the quasi-Laplacian matrix of a graph G and B be the adjacency matrix of the line graph of G, respectively. In this paper, we first present two sharp upper bounds for the largest Laplacian eigenvalue of G by applying the non-negative matrix theory to the similar matrix D −½ KD ½ and U −½ BU ½, respectively, where D is the degree diagonal matrix of G and U = diag(d u d v : uv ε E(G)). And then we give another type of the upper bound in terms of the degree of the vertex and the edge number of G. Moreover, we determine all extremal graphs which achieve these upper bounds. Finally, some examples are given to illustrate that our results are better than the earlier and recent ones in some sense.
Similar content being viewed by others
References
Merris R. Laplacian matrices of graphs: a survey. Linear Algebra Appl, 197–198: 143–176 (1994)
Mohar B. Some applications of Laplace eigenvalues of graphs. In: Harn G, Sabiussi G, eds. Graph Symmetry. Dordrecht: Kluwer Academic Publishers, 1997
Li J S, Zhang X D. A new upper bound for eigenvalues of the Laplacian matrix of a graph. Linear Algebra Appl, 265: 93–100 (1997)
Merris R. A note on Laplacian graph eigenvalues. Linear Algebra Appl, 285: 33–35 (1998)
Li J S, Zhang X D. On the Laplacian eigenvalues of a graph. Linear Algebra Appl, 285: 305–307 (1998)
Rojo O, Soto R, Rojo H. An always nontrivial upper bound for Laplacian graph eigenvalues. Linear Algebra Appl, 312: 155–159 (2000)
Li J S, Pan Y L. De Caen’s inequality and bounds on the largest Laplacian eigenvalue of a graph. Linear Algebra Appl, 328: 153–160 (2001)
Das K C. An improved upper bound for Laplacian graph eigenvalues. Linear Algebra Appl, 368: 269–278 (2003)
Das K C. A characterization on graphs which achieve the upper bound for the largest Laplacian eigenvalue of graphs. Linear Algebra Appl, 376: 173–186 (2004)
Zhang X D. Two sharp upper bounds for the Laplacian eigenvalues. Linear Algebra Appl, 376: 207–213 (2004)
Guo J M. A new upper bound for the Laplacian spectral radius of graphs. Linear Algebra Appl, 400: 61–66 (2005)
Horn R A, Johnson C R. Matrix Analysis. New York: Cambridge University Press, 1985
Li L L. A simplified Brauer’s theorem on matrix eigenvalues. Appl Math J Chinese Univ Ser B, 14(3): 259–264 (1999)
Pan Y L. Sharp upper bounds for the Laplacian graph eigenvalues. Linear Algebra Appl, 355: 287–295 (2002)
Shu J L, Hong Y, Wen R K. A Sharp upper bound on the largest eigenvalue of the Laplacian matrix of a graph. Linear Algebra Appl, 347: 123–129 (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the Natural Science Foundation of Sichuan Province (Grant No. 2006C040)
Rights and permissions
About this article
Cite this article
Wang, Tf. Several sharp upper bounds for the largest laplacian eigenvalue of a graph. Sci. China Ser. A-Math. 50, 1755–1764 (2007). https://doi.org/10.1007/s11425-007-0126-0
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11425-007-0126-0