Soheila KHOJASTEH, Mohammad Javad NIKMEHR

On the nilpotent graph of a ring

Let R be a ring with unity. The nilpotent graph of R, denoted by GN(R), is a graph with vertex set ZN(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(GN(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(GN(Mn(F))) = 2. Also, we determine diam (GN (M2(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.

Anahtar Kelimeler:

Nilpotent graph, diameter, girth

On the nilpotent graph of a ring

Keywords:

Nilpotent graph, diameter, girth,

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Akbari, S., Mohammadian, A.: On zero-divisor graphs of ﬁnite rings, J. Algebra 13, 168–184 (2007).
Akbari, S., Mohammadian, A.: Zero-divisor graphs of non-commutative rings, J. Algebra 296, 462–479 (2006).
Anderson, D.F., Axtell, M.C., Stickles, J.A., Jr.: Zero-divisor graphs in commutative rings, in Commutative Algebra, Noetherian and Non-Noetherian Perspectives (Eds.: M. Fontana, S.-E. Kabbaj, B. Olberding, I. Swanson) Springer-Verlag, New York 23–45 (2011).
Anderson, D.F., Badawi, A.: The total graph of a commutative ring, J. Algebra 320, 2706–2719 (2008).
Anderson, D.F., Livingston, P.S.: The zero-divisor graph of a commutative ring, J. Algebra 217, 434–447 (1999). Anderson, D.F., Mulay, S.B.: On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra 210, 543–550 (2007).
Chen, P.W.: A kind of graph structure of rings, Algebra Colloq. 10:2, 229–238 (2003).
Lam, T.Y.: A First Course in Noncommutative Rings, Springer-Verlag, New York, Inc, 1976.
Li, A.H., Li, Q.H.: A Kind of Graph Structure on Non-reduced Rings, Algebra Colloq. 17:1, 173–180 (2010).
Li, A.H., Li, Q.S.: A kind of graph structure on Von-Neumann Regular Rings, International J. Algebra 4, 291–302 (2010).
Malik, D.S., Mordeson, J.M., Sen, M.K.: Fundamentals of Abstract Algebra, McGraw-Hill, New York, Inc, 1997. Nikmehr, M.J., Shaveisi, F.: The regular digraph of ideals of a commutative ring, Acta Math. Hungar., 134 (4), 516-528 (2012).