Spectral conditions for matching extension

Abstract

A graph G is called k-extendable if for any matching M of size k in G, there exists a perfect matching of G containing M. Let $D\left(G\right)$ and $A\left(G\right)$ be the degree diagonal matrix and the adjacency matrix of G, respectively. For $0\le \alpha <1$, the spectral radius of $A_{\alpha }\left(G\right)=\alpha D\left(G\right)+(1-\alpha )A\left(G\right)$ is called the α-spectral radius of G. In this paper, we give a sufficient condition for a graph G to be k-extendable in terms of the α-spectral radius of G and characterize the corresponding extremal graphs. Moreover, we determine the spectral and signless Laplacian spectral radius conditions for a balanced bipartite graph to be k-extendable.