Signless Laplacian spectral radius and fractional matchings in graphs

A fractional matching of a graph G is a function f giving each edge a number in [0, 1] so that Sigma(e is an element of Gamma(upsilon))f(e) <= 1 for each vertex upsilon is an element of V(G), where Gamma(upsilon) is the set of edges incident to upsilon. The fractional matching number of G, written alpha*'(G), is the maximum value of Sigma(e is an element of Gamma(upsilon))f(e) over all fractional matchings. In this paper, we investigate the relations between the fractional matching number and the signless Laplacian spectral radius of a graph. Moreover, we give some sufficient conditions for the existence of a fractional perfect matching of a graph in terms of the signless Laplacian spectral radius of the graph and its complement. (C) 2020 Elsevier B.V. All rights reserved.