The IP -graph of a naturally valenced association scheme and some of its properties have been studied recently. In this paper we introduce the bipartite version of this graph for a naturally valenced association scheme (X, S), denoted by BIP(S). We also investigate some of its properties. | Turkish Journal of Mathematics Research Article Turk J Math (2013) 37: 560 – 570 ¨ ITAK ˙ c TUB doi: On some results on IP-graphs Roghayeh HAFEZIEH, Mohammadali IRANMANESH∗ Department of Mathematics, Yazd University, 89195-741 Yazd, Iran Received: • Accepted: • Published Online: • Printed: Abstract: The IP -graph of a naturally valenced association scheme and some of its properties have been studied recently. In this paper we introduce the bipartite version of this graph for a naturally valenced association scheme (X, S) , denoted by BIP(S) . We also investigate some of its properties. Key words: Association scheme, common divisor graph, prime graph, bipartite divisor graph 1. Introduction Let G be a group acting transitively on a set X such that all subdegrees are finite. Isaacs and Praeger introduced the concept of the common divisor graph of (G, X) in order to study the relations among all subdegrees of (G, X). They investigated the connectivity of this graph. The main result in [5] deals with the number of connected components of the graph, and the diameter of each connected component. They proved that the common divisor graph of (G, X) has at most two nontrivial components. If (G, X) has only one nontrivial component, then the diameter of that component is at most four, otherwise one of these components is a complete graph and the other has diameter at most two. The common divisor graph of (G, X) is called an IP -graph of (G, X) due to Neumann [7]. The common divisor graph of (G, X) is also studied by Kaplan [6]. Other related research can be found in [1]. Let G be a group acting transitively on a set X such that all subdegrees are finite. Actually this action of G on X induces a naturally valenced association scheme S on X . By the motivation of the common divisor graph of (G, X), Camina [2] introduced the IP -graph of a naturally valenced .
