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On Laplacian Eigenvalues of a Graph Bo Zhou

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On Laplacian Eigenvalues of a Graph

Bo Zhou

Department of Mathematics, South China Normal University, Guangzhou 510631, P.R. China Reprint requests to B. Z.; e-mail: zhoubo@scnu.edu.cn

Z. Naturforsch. 59a, 181 – 184 (2004); received November 11, 2003

Let G be a connected graph with n vertices and m edges. The Laplacian eigenvalues are denoted byµ1(G)≥µ2(G)≥ ··· ≥µn−1(G)>µn(G) =0. The Laplacian eigenvalues have important appli- cations in theoretical chemistry. We present upper bounds forµ1(G) +···+µk(G)and lower bounds forµn−1(G)+···+µn−k(G)in terms of n and m, where 1≤k≤n−2, and characterize the extremal cases. We also discuss a type of upper bounds forµ1(G)in terms of degree and 2-degree.

Key words: Laplacian Eigenvalue; Line Graph; Bipartite Graph.

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