On the mil graph of module over ring

Let M be an unital left module over a ring R with unity. We define an undirected (nil graphs) for the module M as a graph whose vertex set is M^*=M-0 and any two distinct vertices x and y, in these graphs, are adjacent if and only if there exist r ∈ R such that r² (x+y) = 0 and r(x+y)≠ 0. In this paper, we study the graph's adjacency, diameter, radius, and eulerian and hamiltonian properties. We also defined another nil graph Γ*N (M), in which we reduced the vertex set to N(M^*), set of all non-zero nil elements of the module, and keep the adjacency relation same as that of ΓN (M). We investigate the adjacency, diameter, radius, eulerian and hamiltonian properties of the graph Γ*N (Zpⁿ) and compare these properties among both the graphs.