Bipartite graphs with five eigenvalues and pseudo designs.

A pseudo ( v, k, λ)-design is a pair $(X, \mathcal{B})$, where X is a v-set, and $\mathcal{B}=\{B_{1},\ldots,B_{v-1}\}$ is a collection of k-subsets (blocks) of X such that any two distinct B, B intersect in λ elements, and 0≤ λ< k≤ v−1. We use the notion of pseudo designs to characterize graphs of order n whose (adjacency) spectrum contains zero and ± θ with multiplicity ( n−3)/2 where $0<\theta\le\sqrt{2}$. Meanwhile, partial results confirming a conjecture of O. Marrero on a characterization of pseudo ( v, k, λ)-designs are obtained. [ABSTRACT FROM AUTHOR]