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On the Idempotent Graph of Matrix Ring

Avinash Patil, P. S. Momale, C. M. Jadhav

Abstract

Let F be a finite field and R = M2(F) be 2x2 matrix ring over F. In this paper, we explicitly determine all the idempotents in R. Using these idempotents, we study the idempotent graph of R whose vertex set is the set of non-trivial idempotents in R and two idempotents e, f are adjacent if ef = 0 or fe = 0. It is proved that the idempotent graph of R is connected regular graph with diameter 2. Its girth is also characterized. Further, we determine the Wiener and Harary index of the idempotent graph of R.

On the Idempotent Graph of Matrix Ring

Abstract

Let F be a finite field and R = M2(F) be 2x2 matrix ring over F. In this paper, we explicitly determine all the idempotents in R. Using these idempotents, we study the idempotent graph of R whose vertex set is the set of non-trivial idempotents in R and two idempotents e, f are adjacent if ef = 0 or fe = 0. It is proved that the idempotent graph of R is connected regular graph with diameter 2. Its girth is also characterized. Further, we determine the Wiener and Harary index of the idempotent graph of R.
Paper Structure (4 sections, 10 theorems, 7 equations, 1 figure)

This paper contains 4 sections, 10 theorems, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Idempotent in Matrix ring
  3. Variation of Idempotent graph
  4. Wiener and Harary Index

Key Result

Theorem 2.1

Let $R=M_2(\mathbb{F})$. Then $R$ contains exactly $n^2+n+2$ idempotents.

Figures (1)

  • Figure 1: Idempotent graph of $M_2(\mathbb{Z}_2)$ and its variation

Theorems & Definitions (23)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Definition 3.1
  • Example 3.2
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • proof
  • Lemma 3.5
  • ...and 13 more