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 O ϑ ( T ) and the quotient scheme ( X / O ϑ ( T ) , S / / O ϑ ( 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.