Some results on the Laplacian spread of a graph

The Laplacian spread of a graph G with n vertices is defined to be _sL(G)= μ₁(G)-μ _n-1(G), where _μ1(G), _μn-1(G) are the largest and the second smallest Laplacian eigenvalues of G, respectively. It is conjectured that _sL(G)≤n-1. In this paper, we first establish a new sharp upper bound for _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 K₃-free graphs. Finally, we give several sharp lower bounds for _sL(G) as well.