Signless Laplacian spectral radius and fractional matchings in graphs

Abstract A fractional matching of a graph G is a function f giving each edge a number in [ 0 , 1 ] so that ∑ e ∈ Γ ( v ) f ( e ) ≤ 1 for each vertex v ∈ V ( G ) , where Γ ( v ) is the set of edges incident to v . The fractional matching number of G , written α ∗ ′ ( G ) , is the maximum value of ∑ e ∈ E ( G ) 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.