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Domination polynomial of co-maximal graphs of integer modulo ring

Bilal Ahmad Rather

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.

Domination polynomial of co-maximal graphs of integer modulo ring

Abstract

We investigate the domination polynomial of the co-maximal graph related to the ring of integers modulo . Explicit formulas are derived for and , demonstrating that the resulting polynomials exhibit unimodality and log-concavity. For general , we present structural expressions that connect to appropriate induced subgraphs. Finally, we examine domination roots and establish bounds for their moduli using the Eneström--Kakeya theorem.
Paper Structure (3 sections, 9 theorems, 50 equations, 2 figures)

This paper contains 3 sections, 9 theorems, 50 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Domination polynomial of co-maximal graph of $Z_{n}$
  3. Unimodal and log-concave properties of $\Gamma(\mathbb{Z}_{n})$

Key Result

Lemma 2.1

Let $n$ be a positive integer and $d_{i}$ be its proper divisor. Then the following hold.

Figures (2)

  • Figure 1: Block diagram of $\Gamma(\mathbb{Z}_{pqr})$.
  • Figure 2: Representation of zeros of $D(\Gamma(\mathbb{Z}_{2^{5}}),x)$ and $D(\Gamma(\mathbb{Z}_{3\cdot 7}),x)$ on plane.

Theorems & Definitions (9)

  • Lemma 2.1
  • Lemma 2.2: afkhamibanerjeeMatrices
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Proposition 3.1
  • Proposition 3.2