Perfect matchings and Aα-spectral radius in 1-binding graphs

Abstract

Let G be a graph with vertex set V ( G ) and edge set E ( G ) . For α ∈ [ 0,1 ] , we use A α ( G ) and ρ α ( G ) to denote the A α -matrix and the A α -spectral radius of G , respectively. The binding number bind ( G ) of G is defined by bind ( G ) = min | N G ( X ) | | X | : 0̸ ≠ X ⊆ V ( G ) , N G ( X ) ≠ V ( G ) . If bind ( G ) ≥ 1 , then G is called 1-binding. A perfect matching in G is a set of nonadjacent edges covering every vertex of G . Tutte proved that a graph G of even order has a perfect matching if and only if o ( G − S ) ≤ | S | holds for every S ⊆ V ( G ) (Tutte, 1947). In this paper, we use Tutte’s result to prove that a connected 1-binding graph G of even order n with n ≥ n ( α ) has a perfect matching unless G = K 1 ∨ ( K n − 5 ∪ K 3 ∪ K 1 ) if ρ α ( G ) ≥ ρ α ( K 1 ∨ ( K n − 5 ∪ K 3 ∪ K 1 ) ) , where n ( α ) is defined as follows: n ( α ) = max { 18 , 2 + 8 α 1 − 2 α } if α ∈ [ 0 , 1 2 ) , and n ( α ) = 18 if α = 1 2 .