Edge-Disjoint Spanning Trees, Edge Connectivity, and Eigenvalues in Graphs

: Let λ2(G) and τ(G) denote the second largest eigenvalue and the maximum number of edge-disjoint spanning trees of a graph G, respectively. Motivated by a question of Seymour on the relationship between eigenvalues of a graph G and bounds of τ(G), Cioab˘a and Wong conjectured that for any integers d, k ≥ 2 and a d-regular graph G, if λ2(G) < d − 2k−1 d+1 , then τ(G) ≥ k. They proved the conjecture for k = 2, 3, and presented evidence for the cases when k ≥ 4. Thus the conjecture remains open for k ≥ 4. We propose a more general conjecture that for a graph G with Journal of Graph Theory C 2015 Wiley Periodicals, Inc. 16 

• Type: Journal
• Year: 2016
• Where: JGT
• Authors: Xiaofeng Gu, Hong-Jian Lai, Ping Li, Senmei Yao
• Keywords: JGT 2016