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.