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On certain properties of the $p$-unitary Cayley graph over a finite ring

Tung T. Nguyen, Nguyen Duy Tân

Abstract

In recent work, we study certain Cayley graphs associated with a finite commutative ring and their multiplicative subgroups. Among various results that we prove, we provide the necessary and sufficient conditions for such a Cayley graph to be prime. In this paper, we continue this line of research. Specifically, we investigate some basic properties of certain $p$-unitary Cayeley graphs associated with a finite commutative ring. In particular, under some mild conditions, we provide the necessary and sufficient conditions for this graph to be prime.

On certain properties of the $p$-unitary Cayley graph over a finite ring

Abstract

In recent work, we study certain Cayley graphs associated with a finite commutative ring and their multiplicative subgroups. Among various results that we prove, we provide the necessary and sufficient conditions for such a Cayley graph to be prime. In this paper, we continue this line of research. Specifically, we investigate some basic properties of certain -unitary Cayeley graphs associated with a finite commutative ring. In particular, under some mild conditions, we provide the necessary and sufficient conditions for this graph to be prime.
Paper Structure (9 sections, 36 theorems, 73 equations, 3 figures)

This paper contains 9 sections, 36 theorems, 73 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Backgrounds and previous work
  4. $G_R(p)$ where $R$ is a finite local ring of residue characteristics $\ell \neq p$
  5. When is $G_R(p)$ prime
  6. Induced subgraphs of $G_R(p)$
  7. $G_R(p)$ where $p$ is a local ring of residue characteristics $p$
  8. Induced subgraphs of $G_R(p)$
  9. The general case

Key Result

Proposition 2.6

(see chudnovsky2024prime and chudnovsky2024prime) Suppose that ${\rm Cay}(R, S)$ is connected and anti-connected. If ${\rm Cay}(R,S)$ is not prime, then there exists a non-trivial ideal $I$ such that $I$ is a homogeneous set. Furthermore, $I$ is a subset of the Jacobson radical of $R.$

Figures (3)

  • Figure 1: The Cayley graph $G_{\mathop{\mathrm{\mathbb{F}}}\nolimits_{13}}(3)$
  • Figure 2: The Cayley graph $G_{\mathop{\mathrm{\mathbb{F}}}\nolimits_{16}}(5)$
  • Figure 3: The Cayley graph $G_{\mathop{\mathrm{\mathbb{Z}}}\nolimits/25}(5)$

Theorems & Definitions (84)

  • Definition 1.1
  • Remark 1.2
  • Example 1.3
  • Definition 2.1: Induced subgraph
  • Definition 2.2: Tensor product of graphs
  • Definition 2.3: Wreath product
  • Definition 2.4: The complete graph $K_n$
  • Definition 2.5
  • Proposition 2.6
  • Proposition 3.1
  • ...and 74 more