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

Edge connectivity, packing spanning trees, and eigenvalues of graphs

“…Let G ∈ G be a graph shown in Figure 1. By Matlab calculation and Theorems 3.1 and 3.3, we have The following Theorem 3.5 has been given by Gu et al [9]. In the following, we give a different proof by using the signless Laplacian quotient matrix of a graph.…”

“…Lemma 2.3. ( [9]) Let G be a connected graph and π be a partition of V (G). Then δ π ≤ λ 1 (A π (G)) ≤ ∆ π .…”

“…In this section, we will investigate the relationship between other eigenvalues and τ (G), κ ′ (G) in a simple graph G ∈ G. In addition, we know that Gu et al in [9] gave the relationship between µ n−1 , λ 2 and τ (G), κ ′ (G) of a simple graph G, respectively.…”

Spanning tree packing number and eigenvalues of graphs with given girth

“…Several studies made progresses towards Conjecture 1.1, as seen in [5,8,13,14,15]. The conjecture is finally settled in [15].…”

“…Motivated by Kirchhoff's matrix tree theorem [11] and by a problem of Seymour (see Reference [19] of [5]), Cioabȃ and Wong [5] initiated the following conjecture. Conjecture 1.1 (Cioabȃ and Wong [5], Gu et al [8], Li and Shi [13] and Liu et al [14]) Let k be an integer with k ≥ 2 and G be a graph with minimum degree δ ≥ 2k and maximum degree…”