A Characterization of Certain Families of 4-Valent Symmetric Graphs

Let Γ be a connected, 4-valent, G-symmetric graph. Each normal subgroup N of G gives rise to a natural symmetric quotient ΓN, the vertices of which are the N-orbits on V Γ. If this quotient ΓN is not itself 4-valent, then it was shown in [1] that either (i) N has at most two orbits on vertices of Γ, or (ii) N has r ⩾ 3 orbits on vertices and the quotient ΓN is a circuit of length r . In the case in which N is elementary abelian, the graphs which can occur in (i) were classified in [1]. This paper classifies the most symmetrical graphs which can occur in (ii). We show that if N is a minimal normal elementary abelian p-subgroup of G and ΓN is a circuit, then if p = 2, Γ = C (2; r, s), and if p is odd then provided that the stabilizer of a vertex is as large as it can possibly be, Γ must be one of the graphs C(p; r, s), C±1(p; st, s) or C±ε(p; 2st, s) (Theorem 1.1). We also obtain a complete classification in the non-extremal case when p is odd and |N| ⩽ p2. For all other cases we obtain much detailed information about Γ and G, but this does not appear sufficient to allow a general classification of possible pairs (Γ, G).