Non-identity order divisor graphs of groups

Let G be a group with identity e. In this paper, we define and study the non-identity order divisor graph of G, where the vertex set is G-\{e\} and two distinct vertices x and y are adjacent if and only if either O(x)|O(y) or O(y)|O(x). We denote the order divisor graph of group G by \o(G). We study some basic properties of \o(G) such as connectedness, completeness, bipartiteness and Eulerian property. The lower bound as well as the number of edges of \o(G) are also calculated for some group G and some characterizations for fundamental properties of \o(G) have been obtained. Finally, we explore the relation between the order prime graph and the non-identity order divisor graph of some group G.

کلیدواژه ها:

Non-identity order divisor graph ، order prime graph ، eulerian graph ، Finite group