The Order Gcd Graph

Abstract

In this paper we define a simple undirected graph O_g(Z_n) whose vertices are all the elements of Z_n and two distinct vertices a, b are adjacent if and only if gcd (O(a), O(b))= O(a·b) . We introduced that the graph O_g(Z_n)  is complete graph for n=pq . where p, q are distinct prime. The graph O_g(Z_n)  is not planar for n= p² and n= 2^k p. where, p is any odd prime and k > 1 . Also, discuss about Eulerian property of the graph and find degree of vertices and clique number of the graph.