Some spectral conditions for star-factors in bipartite graphs

A spanning subgraph F of G is called an F-factor if every component of F is isomorphic to some member of F, where F is a set of connected graphs. We denote by A(G) the adjacency matrix of G, and by D(G) the distance matrix of G. The largest eigenvalue of A(G), denoted by p(G), is called the adjacency spectral radius of G. The largest eigenvalue of D(G), denoted by mu (G), is called the distance spectral radius of G. In this paper, we aim to provide two spectral conditions to ensure the existence of star-factors with given properties. Let G be a k-edge-connected bipartite graph with bipartition (A, B) and |B|k|A| kn, where n is a sufficiently large positive integer. Then the following two results are true. (i) If p(G) >= rho(Kn-1,kn-k-1VK1,k+1), then G contains a star-factor F with dr(u) = k for any u e A and dr(v) = 1 for any ve B, unless GKn-1.kn-k-1VK1,k+1- (ii) If mu(G) <= mu(Kn-1.kn-k-1VK1,k+1), then G contains a star-factor F with dr(u) = k for any u e A and de(v) = 1 for any ve B, unless GKn-1.kn-k-1VK1,k+1- (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, Al training, and similar technologies.