Anderson, David F.; Livingston, Philip S. The zero-divisor graph of a commutative ring. (English) Zbl 0941.05062 J. Algebra 217, No. 2, 434-447 (1999). The authors study properties of the graph \(\Gamma(R)\) of a commuting ring \(R\) (with \(1\)) defined on the set of nonzero zero-divisors with adjacency relation \((x,y)\in E\) if \(xy= 0\) noting that the class of such graphs is strongly restricted by the (commutative) ring properties of \(R\). They observe that \(\Gamma(R)\) has small diameter \((\leq 3)\) and small girth \((\leq 4)\) among other results. Other classes of algebras (e.g., BCK-algebras) with a \(0\) element permit the same definition, and produce (di)graphs of different and greater variety. An interesting problem from the graph theory point of view is to find a “best class” of algebras which permits any (di)graph \(\Gamma\) to be represented as \(\Gamma(A)\) for some \(A\) in this “best class”. Reviewer: Joseph Neggers (Tuscaloosa) Cited in 23 ReviewsCited in 605 Documents MSC: 05C99 Graph theory 13A99 General commutative ring theory Keywords:commuting ring; zero-divisors; diameter; girth PDF BibTeX XML Cite \textit{D. F. Anderson} and \textit{P. S. Livingston}, J. Algebra 217, No. 2, 434--447 (1999; Zbl 0941.05062) Full Text: DOI Link References: [1] Anderson, D. D.; Naseer, M., Beck’s coloring of a commutative ring, J. Algebra, 159, 500-514 (1993) · Zbl 0798.05067 [2] Atiyah, M. F.; MacDonald, I. G., Introduction to Commutative Algebra (1969), Addison-Wesley: Addison-Wesley Reading · Zbl 0175.03601 [3] Beck, I., Coloring of commutative rings, J. Algebra, 116, 208-226 (1988) · Zbl 0654.13001 [4] Bollabás, B., Graph Theory, An Introductory Course (1979), Springer-Verlag: Springer-Verlag New York [5] Diestel, R., Graph Theory (1997), Springer-Verlag: Springer-Verlag New York [6] Ganesan, N., Properties of rings with a finite number of zero-divisors, Math. Ann., 157, 215-218 (1964) · Zbl 0135.07704 [7] Ganesan, N., Properties of rings with a finite number of zero-divisors, II, Math. Ann., 161, 241-246 (1965) · Zbl 0163.28301 [8] Harary, F., Graph Theory (1972), Addison-Wesley: Addison-Wesley Reading · Zbl 0797.05064 [9] Kaplansky, I., Commutative Rings (1974), Univ. of Chicago Press: Univ. of Chicago Press Chicago · Zbl 0203.34601 [10] Koh, K., On “Properties of rings with a finite number of zero-divisors”, Math. Ann., 171, 79-80 (1967) · Zbl 0153.06201 [12] McDonald, B. R., Finite Rings with Identity (1974), Dekker: Dekker New York · Zbl 0294.16012 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.