Extended Idempotent Divisor Graph of Zn

Associate graph (R) is said to be idempotent divisor graph with vertices set in R*=R-{0} and two non- zero distinct vertices a and b are adjacent if and only if a.b=e, where e an idempotent element non equal 1. In this work we study the extended idempotent divisor graph of Zn is denoted by ((R)) , with vertices set in R* that is for any two distinct vertices a and b are adjacent if there are positive integers t_1,t_2 such that a^(t_1 ). b^(t_2 )=e, where a^(t_1 ), b^(t_2 ) 0 and an idempotent element e not equal 1, and we found the order and the size for some kinds of the idempotent divisor graph of the rings Zn. Also, we found the Hosoya polynomial and the Wiener index for these graphs.