Matching Extendability and Connectivity of Regular Graphs from Eigenvalues

Let G be a graph on even number of vertices. A perfect matching of G is a set of independent edges which cover each vertex of G. For an integer \(t\ge 1\), G is t-extendable if G has a perfect matching, and for any t independent edges, G has a perfect matching which contains these given t edges. The graph G is bi-critical if for any two vertices u and v, the graph \(G-\left\{ u,v\right\} \) contains a perfect matching. Let G be a connected k-regular graph. In this paper, we obtain two sharp sufficient eigenvalue conditions for a connected k-regular graph to be 1-extendable and bi-critical, respectively. Our results are in term of the second largest eigenvalue of the adjacency matrix of G. We also give examples that show that there is no good spectral characterization (in term of the second largest eigenvalue of the adjacency matrix) for 2-extendability of regular graphs in general. Also, for any integer \(1\le \ell <\frac{k+2}{2}\), we obtain an eigenvalue condition for G to be \((\ell +1)\)-connected.