Fractional matching number and spectral radius of nonnegative matrices of graphs

Abstract

ABSTRACT A fractional matching of a graph G is a function f:E(G) → [0, 1] such that for any v ∈ V(G), where E G (v) = {e ∈ E(G): e is incident with v in G}. The fractional matching number of G is is a fractional matching of G}. For any real numbers a ≥ 0 and k ∈ (0, n), it is observed that if n = |V(G)| and , then . We determine a function φ(a, n, δ, k) and show that for a connected graph G with n = |V(G)|, , spectral radius λ 1(G) and complement , each of the following holds. If λ 1 (aD(G) + A(G)) < φ(a, n, δ, k), then If then As applications, we prove a relationship between μ f (G) and λ 1(aD(G) + A(G)) for a graph G. Furthermore, sufficient spectral conditions for a graph to have a fractional perfect matching are also obtained.