Vertex-imprimitive symmetric graphs with exactly one edge between any two distinct blocks

Abstract

A graph $\Gamma$ is called $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on the set of ordered pairs of adjacent vertices. We give a classification of $G$-symmetric graphs $\Gamma$ with $V(\Gamma )$ admitting a nontrivial $G$-invariant partition $\mathcal{B}$ such that there is exactly one edge of $\Gamma$ between any two distinct blocks of $\mathcal{B}$. This is achieved by giving a classification of $(G,2)$-point-transitive and $G$-block-transitive designs $\mathcal{D}$ together with $G$-orbits $\Omega$ on the flag set of $\mathcal{D}$ such that $G_{\sigma ,L}$ is transitive on $L\setminus \{\sigma \}$ and $L\cap N=\{\sigma \}$ for distinct $(\sigma ,L),(\sigma ,N)\in \Omega$, where $G_{\sigma ,L}$ is the setwise stabilizer of $L$ in the stabilizer $G_{\sigma }$ of $\sigma$ in $G$. Along the way we determine all imprimitive blocks of $G_{\sigma }$ on $V\setminus \{\sigma \}$ for every $2$-transitive group $G$ on a set $V$, where $\sigma \in V$.

arXiv commentThis is the final version which will appear in JCT(A). The previous title of this paper was "symmetric spreads of complete graphs"