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On Commuting graphs of triangular rings

Hassan Cheraghpour, Nader M. Ghosseiri, Madineh Jafari, Farnaz Seyfpour

Abstract

Let $R$ be a noncommutative ring with identity. The commuting graph of $R$, denoted by $Γ(R)$, is a graph with vertex set $R \setminus Z(R)$, and two vertices $a$, $b$ are adjacent if $a\neq b$ and $ab=ba$. Let $T=Tr(R)$ be the ring of all $2\times 2$ upper triangular matrices over $R$ and $Γ(T)$ be the commuting graph of $T$. In this article, we find the number of edges, cliques, clique number, and independence number of $Γ(T)$ when $R$ is a finite field. Moreover, we show that for the case when $R= \mathbb{Z}_{n}$ is not a field, $Γ(T)$ is connected with diameter 3. Some useful related results are also obtained, some examples are presented and a question is posed.

On Commuting graphs of triangular rings

Abstract

Let be a noncommutative ring with identity. The commuting graph of , denoted by , is a graph with vertex set , and two vertices , are adjacent if and . Let be the ring of all upper triangular matrices over and be the commuting graph of . In this article, we find the number of edges, cliques, clique number, and independence number of when is a finite field. Moreover, we show that for the case when is not a field, is connected with diameter 3. Some useful related results are also obtained, some examples are presented and a question is posed.
Paper Structure (2 sections, 10 theorems, 56 equations)

This paper contains 2 sections, 10 theorems, 56 equations.

Table of Contents

  1. introduction
  2. Main results

Key Result

Lemma 1.2

Let $G=(V,E)$ be a finite graph. Then

Theorems & Definitions (14)

  • Remark 1.1
  • Lemma 1.2: Handshaking Lamma
  • Lemma 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Proposition 2.5
  • Example 2.6
  • Remark 2.7
  • Corollary 2.8
  • ...and 4 more