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

TL;DR

This paper investigates the domination and total domination polynomials of zero-divisor graphs derived from specific classes of finite commutative rings, expanding understanding of their combinatorial properties.

Contribution

It introduces formulas for domination polynomials of zero-divisor graphs of rings like Z_n with various prime factorizations, a novel analysis in algebraic graph theory.

Findings

Derived explicit domination polynomial formulas for rings Z_n with specific prime factorizations.

Analyzed the structure of zero-divisor graphs for these rings to determine domination properties.

Extended the study of domination polynomials to algebraic structures beyond simple graphs.

Abstract

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 $.