Spanning tree packing number and eigenvalues of graphs with given girth

Abstract

Let τ(G) and κ′(G) denote the spanning tree packing number and the edge-connectivity of a graph G, respectively. Cioabă and Wong (2012) in [5] conjectured an explicit relationship between τ(G) and the second largest adjacency eigenvalue λ2(G) of a regular graph. Gu et al. (2016) in [12] presented a more general conjecture on a simple graph G. This conjecture was proved by Liu et al. (2014) in [21] by showing that for any simple graph G with minimum degree δ≥2k≥4, if λ2(G)<δ−2k−1δ+1, then τ(G)≥k. Similar results involving the algebraic connectivity μn−1(G) and the second largest signless Laplacian eigenvalue q2(G) of a graph G were also obtained. In this paper, we determine a Moore function f(δ,g) for a graph G with minimum degree δ and girth g, and prove that if G is a simple graph of order n with minimum degree δ≥2k≥4 and girth g, then (i) If λ2(G)<δ−2k−1f(δ,g), then τ(G)≥k. (ii) If μn−1(G)>2k−1f(δ,g), then τ(G)≥k. (iii) If q2(G)<2δ−2k−1f(δ,g), then τ(G)≥k. The edge-connectivity analogue results are also obtained. Former results in Gu et al. (2016) [12], Li et al. (2013) [18], Liu et al. (2014) [20] and Liu et al. (2014) [21] are extended.