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On the Comaximal Graph of a Commutative Ring

Published online by Cambridge University Press:  20 November 2018

Karim Samei
Karim Samei*
Affiliation:
Department of Mathematics, Bu Ali Sina University, Hamedan, Iran and School of Mathematics, Institute for Research in Fundamental Science (IPM), Tehran, Iran e-mail: [email protected]
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Abstract

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Let $R$ be a commutative ring with 1. In a 1995 paper in J. Algebra, Sharma and Bhatwadekar defined a graph on $R$, $\Gamma \left( R \right)$, with vertices as elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if $Ra\,+\,Rb\,=\,R$. In this paper, we consider a subgraph ${{\Gamma }_{2}}\left( R \right)$ of $\Gamma \left( R \right)$ that consists of non-unit elements. We investigate the behavior of ${{\Gamma }_{2}}\left( R \right)$ and ${{\Gamma }_{2}}\left( R \right)\backslash \text{J}\left( R \right)$, where $\text{J}\left( R \right)$ is the Jacobson radical of $R$. We associate the ring properties of $R$, the graph properties of ${{\Gamma }_{2}}\left( R \right)$, and the topological properties of $\text{Max}\left( R \right)$. Diameter, girth, cycles and dominating sets are investigated, and algebraic and topological characterizations are given for graphical properties of these graphs.

Keywords

13A99comaximaldiametergirthcyclesdominating set
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 2 , 14 June 2014 , pp. 413 - 423
Copyright
Copyright © Canadian Mathematical Society 2014

Footnotes

This research was in part supported by a grant from IPM (No 89130043).

References

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