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 \(\mu \) of G. The cases in which one of the parameters s, t is less than 2 or \(\mu =r\) are already resolved. It is conjectured in Wang et al. (Linear Algebra Appl 579:302–319, 2019) that if \(s, t\ge 2\) and \(\mu \ne r\), then \(\mu =-2, t=2\) and \(G=\overline{(s+1)K_2}\). For \(\mu =-t\) we verify this conjecture to be true. We further study the case in which \(\mu \ne -t\) and confirm the conjecture provided \(t^2-4\mu ^2t-4\mu ^3=0\). 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.