Partial line graph operator and half-arc-transitive group actions (Q2777533)

The authors develop the concept of partial line graph operator \(P1\) and its inverse operator \(A1\) on graphs of valency 4 with balanced orientation in order to study the structure of transitive permutation groups having a non-self-paired suborbit of length 2 relative to which the corresponding orbital graph is connected, with the emphasis on their vertex stabilizers. The results describe the structure of 4-valent graphs admitting such group actions with large vertex stabilizers and explain why certain Cayley graphs are the focal point in the analysis of the mentioned problem. The authors present a necessary condition for a finite group \(H\) to be a vertex stabilizer of a half-arc-transitive action on a 4-valent graph. A construction of an infinite family of half-arc-transitive group actions with vertex stabilizers isomorphic to the nonabelian dihedral group \(D_8\) is given.