Domination polynomial of co-maximal graphs of integer modulo ring

Abstract: We investigate the domination polynomial of the co-maximal graph $Γ(\mathbb{Z}_n)$ related to the ring of integers modulo $n$. Explicit formulas are derived for ( n = p^{n_1} ) and ( n = p^{n_1}q^{n_2} ), demonstrating that the resulting polynomials exhibit unimodality and log-concavity. For general $n$, we present structural expressions that connect $D(Γ(\mathbb{Z}_n),x)$ to appropriate induced subgraphs. Finally, we examine domination roots and establish bounds for their moduli using the Eneström--Kakeya theorem.