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 $\sigma \in V$.