It is proved that the Cartesian product of an odd cycle with the complete graph on 2 vertices, is determined by the spectrum of the adjacency matrix. We also present some computational results on the spectral characterization of cubic graphs on at most 20 vertices.

We consider signed graphs, i.e., graphs with positive or negative signs on their edges. The notion of signed strongly regular graph is recently defined by the author (Signed strongly regular graphs, Proceeding of 48th Annual Iranian Mathematical Conference, 2017). We construct some families of signed strongly regular graphs with only two distinct eigenvalues. The construction is based on the well-known method known as star complement technique.

We consider signed graphs, i.e, graphs with positive or negative signs on their edges. We determine the admissible parameters for the {5, 6, . . . , 10}-regular signed graphs which have only two distinct eigenvalues. For each obtained parameter we provide some examples of signed graphs having two distinct eigenvalues. It turns out to construction of infinitely many signed graphs of each mentioned valency with only two distinct eigenvalues. We prove that for any k ≥ 5 there are infinitely many connected signed k-regular graphs having maximum eigenvalue √ k. Moreover for each m ≥ 4 we construct a signed 8-regular graph with spectrum [4 m , −2 2m ]. These yield infinite family of k-regular Ramanujan graphs, for each k.

We construct graphs that are cospectral but nonisomorphic with Kneser graphs K(n, k), when n = 3k − 1, k > 2 and for infinitely many other pairs (n, k). We also prove that for 3 ≤ k ≤ n − 3 the Modulo-2 Kneser graph K2(n, k) is not determined by the spectrum.