The Zero-Divisor Graphs Which Are Uniquely Determined By Neighborhoods

Abstract

A nonempty simple connected graph G is called a uniquely determined graph, if distinct vertices of G have distinct neighborhoods. We prove that if R is a commutative ring, then Γ(R) is uniquely determined if and only if either R is a Boolean ring or T(R) is a local ring with x² = 0 for any x ∈ Z(R), where T(R) is the total quotient ring of R. We determine all the corresponding rings with characteristic p for any finite complete graph, and in particular, give all the corresponding rings of K_n if n + 1 = p^q for some primes p, q. Finally, we show that a graph G with more than two vertices has a unique corresponding zero-divisor semigroup if G is a zero-divisor graph of some Boolean ring.

Key Words:

• Boolean ring
• Uniquely determined zero-divisor graph
• Z-local ring
• Zero-divisor semigroup

2000 Mathematics Subject Classification:

• 13A99
• 05C62

ACKNOWLEDGMENTS

This research is supported by National Natural Science Foundation of China (No. 10671122), partially by Collegial Natural Science Research Program of Education Department of Jiangsu Province (No. 07KJD110179) and Natural Science Foundation of Shanghai (No. 06ZR14049).

Notes

Communicated by A. Facchini.