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 $t\leq d-2$. This work extends some classical results of von Baebler [Comment. Math. Helv., 10 (1937), pp. 275–287] and Berge [Théorie des Graphes et Ses Applications, Collection Universitaire de Mathematiques II, Dunod, Paris, 1958] and more recent work of Cioabă, Gregory, and Haemers [J. Combin. Theory Ser. B, 99 (2009), pp. 287–297]. 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.