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The $n$-total graph of an integral domain

Myriam AbiHabib, Ayman Badawi

Abstract

Let $R$ be a finite product of integral domains and $D$ be a union of prime ideals (it is possible that $R$ is just an integral domain). Let $n \geq 1$ be a positive integer. This paper introduces the $n$-total graph of a $(R, D)$. The $n$-total graph of $(R, D)$, denoted by $n-T(R)$, is an undirected simple graph with vertex set $R$, such that two vertices $x, y$ in $R$ are connected by an edge if $x^n + y^n \in D$. In this paper, we study some graph properties and theoretical ring structure.

The $n$-total graph of an integral domain

Abstract

Let be a finite product of integral domains and be a union of prime ideals (it is possible that is just an integral domain). Let be a positive integer. This paper introduces the -total graph of a . The -total graph of , denoted by , is an undirected simple graph with vertex set , such that two vertices in are connected by an edge if . In this paper, we study some graph properties and theoretical ring structure.
Paper Structure (3 sections, 22 theorems, 5 figures)

This paper contains 3 sections, 22 theorems, 5 figures.

Table of Contents

  1. Intrudection
  2. The case where $D$ is an ideal
  3. The case where $D$ is not an ideal of $R$

Key Result

Lemma 2.1

Let $R$ be a finite field with $m$ elements and $d=gcd(n,m-1)$. Then $S_n = \{a^n | a \in R \setminus D\}$ is a cyclic subgroup of $R \setminus D$ and $|S_n| = \frac{m-1}{d}$.

Figures (5)

  • Figure 1: The $3\text{-TG}(\mathbb{F}_4)$
  • Figure 2: The $3\text{-}TG(\mathbb{Z}_7 \setminus D)$
  • Figure 3: The $5$-TG$(\mathbb{F}_9 \setminus D)$
  • Figure 4: The $3$-TG$(\mathbb{Z}_2 \times \mathbb{Z}_2)$
  • Figure 5: The $2$-TG$(\mathbb{Z}_2 \times \mathbb{Z}_3)$

Theorems & Definitions (60)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Example 2.4
  • Example 2.5
  • Theorem 2.6
  • proof
  • Example 2.7
  • Example 2.8
  • ...and 50 more