Fractional spanning tree packing, forest covering and eigenvalues

(2) Suppose that G is a graph with nonincreasing degree sequence d1,d2,…,dn and n≥⌊2st⌋+1. Let β=2st−1⌊2st⌋+1∑i=1⌊2st⌋+1di. Then γ(G)≤s/t, if β≥1, or if 0<β<1, n>⌊2s/t⌋+1+2s−2tβ 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 and Wong on the relationship between eigenvalues and spanning tree packing, and sharpens former results in this area.