A Characterization of Certain Families of 4-Valent Symmetric Graphs

✍ Scribed by A. Gardiner; Cheryl E. Praeger

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 532 KB
• Volume: 15
• Category: Article
• ISSN: 0195-6698
• DOI: 10.1006/eujc.1994.1042

✦ Synopsis

Let (\Gamma) be a connected, 4-valent, (G)-symmetric graph. Each normal subgroup (N) of (G) gives rise to a natural symmetric quotient (\Gamma_{N}), the vertices of which are the (N)-orbits on (V \Gamma). If this quotient (\Gamma_{N}) is not itself 4-valent, then it was shown in [1] that either (i) (N) has at most two orbits on vertices of (\Gamma), or (ii) (N) has (r \geqslant 3) orbits on vertices and the quotient (\Gamma_{N}) 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 (\Gamma_{N}) is a circuit, then if (p=2), (\Gamma=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, (\Gamma) must be one of the graphs (C(p ; r, s), C^{ \pm 1}(p ; s t, s)) or (C^{ \pm \varepsilon}(p ; 2 s t, s)) (Theorem 1.1). We also obtain a complete classification in the non-extremal case when (\rho) is odd and (|N| \leqslant p^{2}). For all other cases we obtain much detailed information about (\Gamma) and (G), but this does not appear sufficient to allow a general classification of possible pairs ((\Gamma, G)).