Valencies and IP-graphs of association schemes

Abstract

Let G be a group acting transitively on a set X such that all subdegrees are finite. Isaacs and Praeger [I.M. Isaacs, C.E. Praeger, Permutation group subdegrees and the common divisor graph, J. Algebra 159 (1993) 158–175] and Kaplan [G. Kaplan, On groups admitting a disconnected common divisor graph, J. Algebra 193 (1997) 616–628] studied the common divisor graph of ( G , X ) . The action of G on X induces an association scheme ( X , S ) . Recently, motivated by the common divisor graph of ( G , X ) , Camina [R. Camina, Schemes and the IP-graph, J. Algebraic Combin. 28 (2008) 271–278] introduced the IP-graph of a naturally valenced association scheme. The common divisor graph of ( G , X ) is the IP-graph of the naturally valenced association scheme ( X , S ) arising from the action of G on X. Under a very strong assumption that all paired valencies are equal, Camina [R. Camina, Schemes and the IP-graph, J. Algebraic Combin. 28 (2008) 271–278] proved that the main results in [I.M. Isaacs, C.E. Praeger, Permutation group subdegrees and the common divisor graph, J. Algebra 159 (1993) 158–175] are also true for the IP-graph of a naturally valenced association scheme with paired valencies equal. However, the association scheme arising from the action of a group acting transitively on a set may not satisfy this assumption. The purpose of this paper is to prove similar results for IP-graphs of naturally valenced association schemes without the assumption that all paired valencies are equal, and hence generalize results in [I.M. Isaacs, C.E. Praeger, Permutation group subdegrees and the common divisor graph, J. Algebra 159 (1993) 158–175; R. Camina, Schemes and the IP-graph, J. Algebraic Combin. 28 (2008) 271–278; G. Kaplan, On groups admitting a disconnected common divisor graph, J. Algebra 193 (1997) 616–628].