A pseudo (v, k, λ)-design is a pair (X, B) where X is a v-set and B = {B1, . . . ,B_{v−1}} is a
collection of k-subsets (blocks) of X such that each two distinct B_i,B_j intersect in λ
elements; and 0 < λ < k < v −1. We use the notion of pseudo designs to characterize graphs
of order n whose spectrum contains either ±1 or ±√2 with multiplicity (n −2)/2 or (n −3)/2. It
turns out that the subdivision of the star K1k is determined by its spectrum if k ̸∈ {ℓ^2 −1 | ℓ ∈
N} ∪ {ℓ^2 −ℓ | ℓ ∈ N}. Meanwhile, partial results confirming a conjecture of O. Marrero on
characterization of pseudo (v, k, λ)-designs are obtained.
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