^{}Faculty of Mathematical sciences, Shahrood University of Technology, Shahrood, Iran.

Abstract

Let $R$ be an associative ring with identity. A ring $R$ is called reversible if $ab=0$, then $ba=0$ for $a,b\in R$. The quasi-zero-divisor graph of $R$, denoted by $\Gamma^*(R)$ is an undirected graph with all nonzero zero-divisors of $R$ as vertex set and two distinct vertices $x$ and $y$ are adjacent if and only if there exists $0\neq r\in R \setminus (\mathrm{ann}(x) \cup \mathrm{ann}(y))$ such that $xry=0$ or $yrx=0$. In this paper, we determine the diameter and girth of $\Gamma^*(R)$. We show that the zero-divisor graph of $R$ denoted by $\Gamma(R)$, is an induced subgraph of $\Gamma^*(R)$. Also, we investigate when $\Gamma^*(R)$ is identical to $\Gamma(R)$. Moreover, for a reversible ring $R$, we study the diameter and girth of $\Gamma^*(R[x])$ and we investigate when $\Gamma^*(R[x])$ is identical to $\Gamma(R[x])$.

[1] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999) 434-447. [2] D.F. Anderson and S.B. Mulay, On the diameter and girh of a zero-divisor graph, J. Pure Appl. Algebra 210 (2007) 543-550. [3] A. Badawi, On the annihilator graph of a commutative ring, Commun. Algebra 42 (2014) 1-14. [4] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988) 208-226. [5] V. Camillo and P.P. Nielsen, McCoy rings and zero-divisors, J. Pure Appl. Algebra 212 (2008) 599-615. [6] E. Hashemi, R. Amirjan and A. Alhevaz, On zero-divisor graphs of skew polynomial rings over non-commutative rings, J. Algebra Appl. 16 (2017) 1750056 (14 pages). [7] E. Hashemi and R. Amirjan, Zero-divisor graphs of Ore extensions over reversible rings, Canad. Math. Bull. 59 (2016) 794-805 [8] C.Y. Hong, N.K. Kim, Y. Lee and S.J. Ryu, Rings with Property (A) and their extensions, J. Algebra 315 (2007) 612-628 [9] N.K. Kim and Y. Lee, Extention of reversible rings, J. Pure Appl. Algebra 185 (2003) 207-223. [10] T.Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, Berlin-Heidelberg-New York, 1991. [11] N.H. McCoy, Remarks on divisors of zero, Amer. Math. Monthly 49 (1942) 286-295. [12] P.P. Nielsen, Semi-commutativity and the McCoy condition, J. Algebra 298 (2006) 134-141. [13] S.P. Redmond, The zero-divisor graph of a non-commutative ring, Int. J. Commut. Rings 1 (2002) 203-211. [14] D.B. West, Introduction to Graph Theory, Second Edition, Prentice Hall, Upper Saddle River, 2001. [15] S. Zhao, J. Nan and G. Tang, Quasi-zero-divisor graphs of non-commutative rings, J. Math. Res. Appl. 37 (2017) 137-147.