Weakly central-vertex complete graphs with applications to commutative rings

The zero-divisor graph of a commutative ring RR is the graph whose vertices consist of the nonzero zero-divisors of RR such that distinct vertices xx and yy are adjacent if and only if xy=0xy=0. In this paper, a decomposition theorem is provided to describe weakly central-vertex complete graphs of radius 11. 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)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 11 is provided.