In this paper, we consider a local extension R of the Galois ring of the form GR(p n , d)[x]/(f (x) a ) , where n, d , and a are positive integers; p is a prime; and f (x) is a monic polynomial in GR(p n , d) [x] of degree r such that the reduction f (x) in F p d [x] is irreducible. We establish the exponent of R without complete determination of its unit group structure. We obtain better analysis of the iteration graphs G (k) (R) induced from the k th power mapping including the conditions on symmetric digraphs. In addition, we work on the digraph over a finite chain ring R . The structure of G (k) 2 (R) such as indeg k 0 and maximum distance for G (k) 2 (R) are determined by the nilpotency of maximal ideal M of R .