The zero-divisor graph of a commutative ring R is the graph whose vertices consist of the nonzero zero-divisors of R such that distinct vertices x and y are adjacent if and only if x y = 0. In this paper, a decomposition theorem is provided to describe weakly central-vertex complete graphs of radius 1. This characterization is then applied to the class of zero-divisor graphs of commutative rings. For finite commutative rings whose zero-divisor graphs are not isomorphic to that of Z4 [X] / (X2), it is shown that weak central-vertex completeness is equivalent to the annihilator condition. Furthermore, a schema for describing zero-divisor graphs of radius 1 is provided. © 2009 Elsevier B.V. All rights reserved.
LaGrange, J. D. (2010). Weakly central-vertex complete graphs with applications to commutative rings. Journal of Pure and Applied Algebra, 214(7), 1121–1130. https://doi.org/10.1016/j.jpaa.2009.09.018