Domination polynomial and total domination polynomial of zero-divisor graphs of commutative rings

The domination polynomial (the total domination polynomial) of a graph $G$ of order $n$ is the generating function of the number of dominating sets (total dominating sets) of $G$ of any size. In this paper, we study the domination polynomial and the total domination polynomial of zero-divisor graphs of the ring ${\mathbb{Z}}_n$ where $n\in \{2p,p^2,pq,p^{2}q,pqr,p^{\alpha }\}$, and $p,q,r$ are primes with $p>q>r>2$.