Edge-disjoint spanning trees and eigenvalues

Let tau(G) and lambda(2)(G) be the maximum number of edge-disjoint spanning trees and the second largest eigenvalue of a graph G, respectively. Motivated by a question of Seymour on the relationship between eigenvalues of a graph G and tau(G), Cioaba and Wong conjectured that for any integers k >= 2, d >= 2k and a d-regular graph G, if lambda(2)(G) < d-2k-1/d+1, then tau(G) >= k. They proved this conjecture for k = 2, 3. Gu, Lai, Li and Yao generalized this conjecture to simple graph and conjectured that for any integer k >= 2 and a graph G with minimum degree delta and maximum degree Delta, if lambda(2)(G) < 2 delta - Delta - 2k-1/delta+1 then tau(G) >= k. In this paper, we prove that lambda(2)(G) delta - 2k-2/k/delta+1 implies tau(G) >= k and show the two conjectures hold for sufficiently large n. (C) 2013 Elsevier Inc. All rights reserved.