Document Type : Research Paper


Payame Noor University


Let $R$ be a ring with unity. The undirected nilpotent graph of $R$, denoted by $\Gamma_N(R)$, is a graph with vertex set ~$Z_N(R)^* = \{0\neq x \in R |\ xy \in N(R) \ for\ some\ y \in R^*\}$, and two
distinct vertices $x$ and $y$ are adjacent if and only if $xy \in N(R)$, or equivalently, $yx \in N(R)$, where $N(R)$ denoted the nilpotent elements of $R$. Recently, it has been proved that if $R$ is a left Artinian ring, then $\diam(\Gamma_{N}(R))\leqslant 3$. In this paper, using the concept of rank over commutative rings, we investigate basic properties of undirected nilpotent graph of matrix algebras. Moreover, some result on undirected nilpotent graph of matrix algebras over commutative rings are given. For
instance, we prove that $\Gamma_{N}(M_{n}(R))$ is not planar for all $n\geqslant 2$. Furthermore, we show that $\diam(\Gamma_{N}(R))\leqslant \diam(\Gamma_{N}(M_{n}(R)))$ for an
Artinian commutative ring $R$. Also, we prove that $\Gamma_{N}(M_{n}(R))\cong\Gamma_{N}(M_{n}(T(R)))$, where $T(R)$ be the total quotient ring of a commutative ring $R$


[1] S. Akbari, H.R. Maimani and S. Yassemi, When a zero-divisor graph is planar or a complete r-partite graph, J. Algebra 270, 169-180 (2003).
[2] S. Akbari, A. Mohammadian, Zero-divisor graphs of non-commutative rings, J. Algebra 296, 462{479 (2006). [3] D. F. Anderson, A. Frazier, A. Lauve, P. Livingston, The zero-divisor graph of a commutative ring. II Lect. Notes Pure and Appl. Math 220, 61{72 (2001).
[4] D. F. Anderson, R. Levy, J. Shapiro, Zero-divisor graphs, Von-Neumann regular rings and Boolean algebras, J. Pure Appl. Algebra 180, 221{241 (2003).
[5] D. F. Anderson and P. Livingston. The zero-divisor graph of a commutative ring, J. Algebra 217, 434{447 (1999).
[6] D. F. Anderson, S. B. Mulay,On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra 210, 543-550(2007).
[7] I. Beck, Coloring of commutative rings. J. Algebra 116, 208{226 (1988).
[8] B. Fine, Classi cation of nite rings of order p2, Mathematics Magazine, VOL. 66, NO. 4, 246{252 (1993)
[9] J.A. Bondy, U.S.R. Murty,Graph Theory with Applications, American Elsevier, New York, 1976.
[10] I. Bozic, Z. Petrovic,Zero-divisor graphs of matrices over commutative rings, Communication in Algebra 37, 1186{
1192 (2009)
[11] W. C. Brown, Matrices Over Commutative Rings, Marcel Dekker, Inc, New York, Basel, Hong Kong.
[12] P. W. Chen, A kind of graph structure of rings, Algebra Colloq 10(2), 229{238 (2003).
[13] T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, New York, Inc, 1976.
[14] A. H. Li,Q. H. Li, A kind of graph structure on non-reduced rings, Algebra Colloq 17(1), 173{180 (2010).
[15] A. H. Li, Q. S. Li, A kind of graph structure on Von-Neumann regular rings, International J. Algebra 4, 291-302 (2010).