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), Cioabă 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 EDGE-DISJOINT SPANNING TREES, EDGE CONNECTIVITY 17 minimum degree δ ≥ 2k ≥ 4, if λ2(G) < δ − 2k−1 δ+1 , then τ (G) ≥ k. In this article, we prove that for a graph G with minimum degree δ, each of the following holds. (i) For k ∈ {2,3}, if δ ≥ 2k and λ2(G) < δ − 2k−1 δ+1 , then τ (G) ≥ k. (ii) For k ≥ 4, if δ ≥ 2k and λ2(G) < δ − 3k−1 δ+1 , then τ (G) ≥ k. Our results sharpen theorems of Cioabă and Wong and give a partial solution to Cioabă and Wong’s conjecture and Seymour’s problem. We also prove that for a graph G with minimum degree δ ≥ k ≥ 2, if λ2(G) < δ − 2(k−1) δ+1 , then the edge connectivity is at least k, which generalizes a former result of Cioabă. As corollaries, we investigate the Laplacian and signless Laplacian eigenvalue conditions on τ (G) and edge connectivity. C © 2015 Wiley Periodicals, Inc. J. Graph Theory 81: 16–29, 2016