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 $\overline{f}(x)$ in $\mathbb{F}_{p^{d}}[x]$ 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 $R$.