The Aα-spectral radius and rainbow k-factor

We say that a family of graphs G admits a rainbow k -factor if there exists a k -regular spanning subgraph H of the common vertex set V such that each edge of H is drawn from a distinct graph in G . In this paper, we derive several spectral sufficient conditions under which a family of graphs admits a rainbow substructure. Specifically, let G = { G 1 , G 2 , … , G s + 1 } be a collection of graphs sharing the same vertex set V = { 1,2 , … , n } , and let ρ α ( G i ) denote the spectral radius of the A α -matrix of G i , where A α ( G i ) = α D ( G i ) + ( 1 − α ) A ( G i ) for α ∈ [ 0,1 ) . First, we prove that if, for all i ∈ { 1,2 , … , s + 1 } and for any α ∈ [ 0,1 ) , ρ α ( G i ) ≥ max 2 s , n α + s − 1 + ( n α + s − 1 ) 2 − 4 s α ( s − 1 ) + 4 s ( n − s ) ( 1 − 2 α ) 2 , then G contains a rainbow matching, i.e., a set of s + 1 pairwise disjoint edges { e 1 , e 2 , … , e s + 1 } with e i ∈ E ( G i ) , unless all G i are identical and each equals either K 2 s + 1 ∪ ( n − 2 s − 1 ) K 1 or K s ∨ ( n − s ) K 1 . Second, we establish a spectral sufficient condition — expressed in terms of the A α -spectral radii of the constituent graphs — that guarantees the existence of a rainbow k -factor in G .