On IP-graphs of association schemes and applications to group theory

Let G be a group acting transitively on a set X such that all subdegrees are finite. Isaacs and Praeger (1993) [5], studied the common divisor graph of (G,X). For a group G and its subgroup A, based on the results in Isaacs and Praeger (1993) [5], , Kaplan (1997) [6], proved that if A is stable in G and the common divisor graph of (A,G) has two components, then G has a nice structure. Motivated by the notion of the common divisor graph of (G,X), Camina (2008) [3], introduced the concept of the IP-graph of a naturally valenced association scheme. The common divisor graph of (G,X) is the IP-graph of the association scheme arising from the action of G on X. Xu (2009) [8], studied the properties of the IP-graph of an arbitrary naturally valenced association scheme, and generalized the main results in Isaacs and Praeger (1993) [5], and Camina (2008) [3], . In this paper we first prove that if the IP-graph of a naturally valenced association scheme (X,S) is stable and has two components (not including the trivial component whose only vertex is 1), then S has a closed subset T such that the thin residue Oϑ(T) and the quotient scheme (X/Oϑ(T),S//Oϑ(T)) have very nice properties. Then for an association scheme (X,S) and a closed subset T of S such that S//T is an association scheme on X/T, we study the relations between the closed subsets of S and those of S//T. Applying these results to schurian schemes and common divisor graphs of groups, we obtain the results of Kaplan [6] as direct consequences.