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)−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.