Domatic polynomials of $\Gamma(Z(R)):$ the zero-divisor graphs of commutative rings
Resumo
The domatic polynomial $DP(G, x)$ of a graph $G$ is defined as $DP(G,x)= \sum_{j=1}^{d(G)} dp(G,j) x^j$, where $dp(G,j)$ represents the number of domatic partition of $G$ with size $j$. In this paper, we find domatic number and domatic polynomial of $\Gamma(Z_n)$ where $n \in \{2s, s^2, st, s^2t,stu,s^\alpha\}$ for distinct prime numbers $s, t\; \text{and}\; u$ with $\alpha >2$ and their roots. Further, we discuss a characterization on $DP(\Gamma(R),x)$. Finally, we establish that their domatic polynomials possess the properties of log-concavity and unimodality.
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