On joins of a clique and a co-clique as star complements in regular graphs

In this paper we consider r -regular graphs G that admit the vertex set partition such that one of the induced subgraphs is the join of an s -vertex clique and a t -vertex co-clique and represents a star complement for an eigenvalue μ of G . The cases in which one of the parameters s , t is less than 2 or μ = r are already resolved. It is conjectured in Wang et al. (Linear Algebra Appl 579:302–319, 2019) that if s , t ≥ 2 and μ ≠ r , then μ = - 2 , t = 2 and G = ( s + 1 ) K 2 ¯ . For μ = - t we verify this conjecture to be true. We further study the case in which μ ≠ - t and confirm the conjecture provided t 2 - 4 μ 2 t - 4 μ 3 = 0 . For the remaining possibility we determine the structure of a putative counterexample and relate its existence to the existence of a particular 2-class block design. It occurs that the smallest counterexample would have 1265 vertices.