Abstract. The famous Erdős–Sós conjecture states that every graph of average degree more than [Formula: see text] must contain every tree on [Formula: see text] vertices. In this paper, we study a spectral version of this conjecture. For [Formula: see text], let [Formula: see text] be the join of a clique on [Formula: see text] vertices with an independent set of [Formula: see text] vertices and denote by [Formula: see text] the graph obtained from [Formula: see text] by adding one edge. We show that for fixed [Formula: see text] and sufficiently large [Formula: see text], if a graph on [Formula: see text] vertices has adjacency spectral radius at least as large as [Formula: see text] and is not isomorphic to [Formula: see text], then it contains all trees on [Formula: see text] vertices. Similarly, if a sufficiently large graph has spectral radius at least as large as [Formula: see text], then it either contains all trees on [Formula: see text] vertices or is isomorphic to [Formula: see text]. This answers a two-part conjecture of Nikiforov affirmatively.

We determine all graphs for which the adjacency matrix has at most two eigenvalues (multiplicities included) not equal to $$-2$$ , or 0, and determine which of these graphs are determined by their adjacency spectrum.

From Alon and Boppana, and Serre, we know that for any given integer $k\geq 3$ and real number $\lambda<2\sqrt{k-1}$, there are only finitely many $k$-regular graphs whose second largest eigenvalue is at most $\lambda$. In this paper, we investigate the largest number of vertices of such 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 $t\leq d-2$. 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.