The order difference interval graph of a group

In this paper we introduce the concept of order difference interval graph ¡ODI (G) of a group G. It is a graph ¡ODI (G) with V (¡ODI (G)) = G and two vertices a and b are adjacent in ¡ODI (G) if and only if o (b) − o (a) ∈ [o (a)، o (b)]. Without loss of generality، we assume that o (a) ≤ o (b). In this paper we obtain several properties of ¡ODI (G)، upper bounds on the number of edges of ¡ODI (G) and determine those groups whose order difference interval graph is isomorphic to a complete multipartite graph.