Abstract: A fractional matching of a graph G is a function f:E(G) →[0, 1] such that for each vertex v, Σ_e∈Γ_G(v) f (e) ≤ 1. The fractional matching number of G is the maximum value of Σ_e∈E(G) f (e) over all fractional matchings f. Tian et al. (Linear Algebra Appl 506:579-587, 2016) determined the extremal graphs with minimum distance Laplacian spectral radius among n-vertex graphs with given matching number. However, a natural problem is left open:among all n-vertex graphs with given fractional matching number, how about the lower bound of their distance Laplacian spectral radii and which graphs minimize the distance Laplacian spectral radii? In this paper, we solve these problems completely.

Key words: Distance Laplacian, Spectral radius, Fractional matching number