On the Idempotent Graph of Matrix Ring

TL;DR

This paper explicitly characterizes all idempotents in the 2x2 matrix ring over a finite field, studies the properties of their associated graph, and computes key graph invariants.

Contribution

It provides a complete description of the idempotent graph of M2(F), including its connectivity, regularity, diameter, girth, and indices, which was not previously known.

Findings

The idempotent graph is connected and regular with diameter 2.

The girth of the idempotent graph is characterized.

The Wiener and Harary indices of the graph are computed.

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.