# Turkish Journal of Mathematics

## DOI

10.3906/mat-1112-35

## Abstract

Let R be a ring with unity. The nilpotent graph of R, denoted by \Gamma_N(R), is a graph with vertex set Z_N(R)^* = {0 \neq x \in R \mid xy \in N(R) for some 0 \neq y \in R}; and two distinct vertices x and y are adjacent if and only if xy \in N(R), where N(R) is the set of all nilpotent elements of R. Recently, it has been proved that if R is a left Artinian ring, then diam(\Gamma_N(R)) \leq 3. In this paper, we present a new proof for the above result, where R is a finite ring. We study the diameter and the girth of matrix algebras. We prove that if F is a field and n \geq 3, then diam(\Gamma_N(M_n(F))) = 2. Also, we determine diam (\Gamma_N (M_2(F))) and classify all finite rings whose nilpotent graphs have diameter at most 3. Finally, we determine the girth of the nilpotent graph of matrix algebras.

## Keywords

Nilpotent graph, diameter, girth

## First Page

553

## Last Page

559

## Recommended Citation

NIKMEHR, MOHAMMAD JAVAD and Khojasteh, Soheila
(2013)
"On the nilpotent graph of a ring,"
*Turkish Journal of Mathematics*: Vol. 37:
No.
4, Article 1.
https://doi.org/10.3906/mat-1112-35

Available at:
https://journals.tubitak.gov.tr/math/vol37/iss4/1