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

Abstract

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) K2¯. For μ= - t we verify this conjecture to be true. We further study the case in which μ≠ - t and confirm the conjecture provided t2- 4 μ2t- 4 μ3= 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.