On the total graph of a ring and its related graphs: a survey. (English) Zbl 1327.13007
Fontana, Marco (ed.) et al., Commutative algebra. Recent advances in commutative rings, integer-valued polynomials, and polynomial functions. Based on mini-courses and a conference on commutative rings, integer-valued polynomials and polynomial functions, Graz, Austria, December 16–18 and December 19–22, 2012. New York, NY: Springer (ISBN 978-1-4939-0924-7/hbk; 978-1-4939-0925-4/ebook). 39-54 (2014).
Summary: Let \(R\) be a (commutative) ring with nonzero identity and \(Z(R)\) be the set of all zero divisors of \(R\). The total graph of \(R\) is the simple undirected graph \(T(\Gamma(R))\) with vertices all elements of \(R\), and two distinct vertices \(x\) and \(y\) are adjacent if and only if \(x+y\in Z(R)\). This type of graphs has been studied by many authors. In this paper, we state many of the main results on the total graph of a ring and its related graphs.
For the entire collection see [Zbl 1294.13002].
For the entire collection see [Zbl 1294.13002].
MSC:
| 13A15 | Ideals and multiplicative ideal theory in commutative rings |
| 13B99 | Commutative ring extensions and related topics |
| 05C99 | Graph theory |
Keywords:
total graph; zero divisors; diameter; girth; connected graph; genus; generalized total graph; dominating set; clique; chromatic numberReferences:
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