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 $n\in\lbrace 2p, p^2, pq, p^2q, pqr, p^\alpha\rbrace$, and $p, q, r$ are primes with $p>q>r>2$.