Two spectral characterizations of regular, bipartite graphs with five eigenvalues

Graphs with a few distinct eigenvalues usually possess an interesting combinatorial structure. We show that regular, bipartite graphs with at most six distinct eigenvalues have the property that each vertex belongs to the constant number of quadrangles. This enables to determine, from the spectrum alone, the feasible families of numbers of common neighbors for each vertex with other vertices in its part. For particular spectra, such as [6, 2, 0,−2,−6] (where exponents denote eigenvalue multiplicities), there is a unique such family, which makes it possible to characterize all graphs with this spectrum. Using this lemma we also to show that, for r ≥ 2, a graph has spectrum [r, √ r r(r−1) , 02(r−1),− √ r r(r−1) ,−r] if and only if it is a graph of a 1-resolvable transversal design TD(r, r), i.e., if it corresponds to the complete set of mutually orthogonal Latin squares of size r in a well-defined manner.