Abstract
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 [Theorie des Graphes et Ses Applications, Collection Universitaire de Mathematiques II, Dunod, Paris, 1958] and more recent work of Cioaba, 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.
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