Article ID Journal Published Year Pages File Type
4598533 Linear Algebra and its Applications 2016 16 Pages PDF
Abstract

The Laplacian spread of a graph G with n   vertices is defined to be sL(G)=μ1(G)−μn−1(G)sL(G)=μ1(G)−μn−1(G), where μ1(G)μ1(G), μn−1(G)μn−1(G) are the largest and the second smallest Laplacian eigenvalues of G  , respectively. It is conjectured that sL(G)≤n−1sL(G)≤n−1. In this paper, we first establish a new sharp upper bound for sL(G)sL(G), and then use it to prove that the conjecture is true for t  -quasi-regular graphs when t≤n−3+2/n. We also present some other partial solutions for this conjecture; in particular, we show that the conjecture holds for K3K3-free graphs. Finally, we give several sharp lower bounds for sL(G)sL(G) as well.

Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory
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