Classifying a family of symmetric graphs (Q2721599)

A finite graph \(\Gamma\) is said to be a \(G\)-symmetric graph if it admits a finite group \(G\) as group of automorphisms such that \(G\) is transitive on the vertices of \(\Gamma\) and, in its induced action, is transitive on the arcs of \(\Gamma\). Let \(\Gamma\) be a \(G\)-symmetric graph admitting a nontrivial \(G\)-invariant partition \({\mathfrak B}\) of block size \(v\). For blocks \(B\), \(C\) of \({\mathfrak B}\) adjacent in the quotient \(\Gamma_{\mathfrak B}\), let \(k\) be the number of vertices in \(B\) adjacent to at least one vertex in \(C\). In the present paper possibilities for \((\Gamma,\Gamma_{\mathfrak B},G)\) with \(k=v-1\geq 2\) and \({\mathfrak B}(\alpha)={\mathfrak B}(\beta)\) for adjacent vertices \(\alpha\), \(\beta\) of \(\Gamma\) are classified, where \({\mathfrak B}(\gamma)\) denotes the set of blocks \(C\) such that \(\gamma\) is the only vertex in \(B\) not adjacent to any vertex in \(C\).