Edge-Connectivity, Eigenvalues, and Matchings in Regular Graphs

In this paper, we study the relationship between eigenvalues and the existence of certain subgraphs in regular graphs. We give a condition on an appropriate eigenvalue that guarantees a lower bound for the matching number of a t-edge-connected d-regular graph, when t ≤ d − 2. This work extends some classical results of von Baebler and Berge and more recent work of Cioabă, Gregory, and Haemers. We also study the relationships between the eigenvalues of a d-regular t-edge-connected graph G and the maximum number of pairwise disjoint connected subgraphs in G that are each joined to the rest of the graph by exactly t edges.