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.