Finite symmetric graphs with two-arc transitive quotients (Q1775894)

An \(s\)-arc of a finite graph \(X\) is a sequence \((v_0,v_1,\dots,v_s)\) of \(s+1\) vertices of \(X\) such that \(v_iv_{i+1}\), \(i=0,1,\dots,s-1\), is an edge of \(X\), and \(v_i\neq v_{i+2}\), \(i=0,1,\dots,i-2\). This fine paper is part of series studying graphs \(X\) that admit a group \(G\) of automorphisms acting transitively on the 2-arcs of \(X\). The 1-arcs of \(X\) are simply called {arcs}. The graph \(X\) is \(G\)-symmetric if the group \(G\) of automorphisms of \(X\) acts transitively on the arcs of \(X\). Further, for such a group \(G\), a partition \(\mathcal{B}\) of the vertex set of \(X\) is \(G\)-invariant if \(G\) preserves the partition. If the partition \(\mathcal{B}\) is \(G\)-invariant, each part in \(\mathcal{B}\) must have the same cardinality. The partition \(\mathcal{B}\) is {nontrivial} as long as the parts are neither singletons nor all of \(V(X)\). Given a nontrivial partition \(\mathcal{B}\) of \(X\), there is a natural quotient graph \(X_{\mathcal{B}}\) that arises. The authors study 2-arc transitivity of \(X_{\mathcal{B}}\) for a \(G\)-invariant partition \(X_{\mathcal{B}}\), where \(X\) is \(G\)-symmetric.