LAUSR

Abstract A perfect matching in a graph G is a set of disjoint edges covering all vertices of G. Let ρ ( G ) be the spectral radius of a graph G, and let θ ( n ) be the largest root of x 3 − ( n − 4 ) x 2 − ( n − 1 ) x + 2 ( n − 4 ) = 0 . In this paper, we prove that for a positive even integer n ≥ 8 or n = 4 , if G is an n-vertex graph with ρ ( G ) > θ ( n ) , then G has a perfect matching; for n = 6 , if ρ ( G ) > 1 + 33 2 , then G has a perfect matching. It is sharp for every positive even integer n ≥ 4 in the sense that there are graphs H with ρ ( H ) = θ ′ ( n ) and no perfect matching, where θ ′ ( n ) = θ ( n ) if n = 4 or n ≥ 8 and θ ′ ( 6 ) = 1 + 33 2 .