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Paper
Title

# Nilpotent graphs of skew polynomial rings over non-commutative rings

Year: 1399
COI: JR_COMB-9-1_004
Language: EnglishView: 26
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### Authors

Mohammad Javad Nikmehr - K.N.Toosi University
Abdolreza Azadi - ‎K‎. ‎N‎. ‎Toosi University of Technology

### Abstract:

Let $R$ be a ring and $\alpha$ be a ring endomorphism of $R$‎. ‎The undirected nilpotent graph of $R$‎, ‎denoted by $\Gamma_N(R)$‎, ‎is a graph with vertex set $Z_N(R)^*$‎, ‎and two distinct vertices $x$ and $y$ are connected by an edge if and only if $xy$ is nilpotent‎, ‎where $Z_N(R)=\{x\in R\;|\; xy\; \rm{is\; nilpotent,\;for\; some}\; y\in R^*\}.$ In this article‎, ‎we investigate the interplay between the ring theoretical properties of a skew polynomial ring $R[x;\alpha]$ and the graph-theoretical properties of its nilpotent graph $\Gamma_N(R[x;\alpha])$‎. ‎It is shown that if $R$ is a symmetric and $\alpha$-compatible with exactly two minimal primes‎, ‎then $diam(\Gamma_N(R[x,\alpha]))=۲$‎. ‎Also we prove that $\Gamma_N(R)$ is a complete graph if and only if $R$ is isomorphic to $\Z_۲\times\Z_۲$‎.

### Paper COI Code

This Paper COI Code is JR_COMB-9-1_004. Also You can use the following address to link to this article. This link is permanent and is used as an article registration confirmation in the Civilica reference:

https://civilica.com/doc/1194854/

### How to Cite to This Paper:

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Nikmehr, Mohammad Javad and Azadi, Abdolreza,1399,Nilpotent graphs of skew polynomial rings over non-commutative rings,https://civilica.com/doc/1194854

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## Scientometrics

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Type of center: دانشگاه دولتی
Paper count: 10,706
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