Fractional spanning tree packing, forest covering and eigenvalues

We investigate the relationship between the eigenvalues of a graph G and fractional spanning tree packing and coverings of G. Let ω(G) denote the number of components of a graph G. The strength η(G) and the fractional arboricity γ (G) are defined by η(G) = min |X | ω(G − X) − ω(G) , and γ (G) = max |E(H)| |V (H)| − 1 , where the optima are taken over all edge subsets X whenever the denominator is non-zero. The well known spanning tree packing theorem by Nash-Williams and Tutte indicates that a graph G has k edge-disjoint spanning tree if and only if η(G) ≥ k; and Nash-Williams proved that a graph G can be covered by at most k forests if and only if γ (G) ≤ k. Let λi(G) (μi(G), qi(G), respectively) denote the ith largest adjacency (Laplacian, signless Laplacian, respectively) eigenvalue of G. In this paper, we prove the following. (1) Let G be a graph with δ ≥ 2s/t . Then η(G) ≥ s/t if μn−1(G) > 2s−1 t(δ+1) , or if λ2(G) < δ − 2s−1 t(δ+1) , or if q2(G) < 2δ − 2s−1 t(δ+1) . (2) Suppose that G is a graph with nonincreasing degree sequence d1, d2, . . . , dn and n ≥ ⌊ 2s t ⌋ + 1. Let β = 2s t − 1 ⌊ 2s t ⌋+1 ⌊ 2s t ⌋+1 i=1 di. Then γ (G) ≤ s/t , if β ≥ 1, or if 0 < β < 1, n > ⌊2s/t⌋ + 1 + 2s−2 tβ and μn−1(G) > n(2s/t − 2/t − β(⌊2s/t⌋ + 1)) (⌊2s/t⌋ + 1)(n − ⌊2s/t⌋ − 1) . Our result proves a stronger version of a conjecture by Cioabă andWong on the relationship between eigenvalues and spanning tree packing, and sharpens former results in this area. © 2016 Elsevier B.V. All rights reserved. ∗ Corresponding author. E-mail address: [email protected] (X. Gu). http://dx.doi.org/10.1016/j.dam.2016.04.027 0166-218X/© 2016 Elsevier B.V. All rights reserved. 220 Y. Hong et al. / Discrete Applied Mathematics 213 (2016) 219–223