The Zero-Divisor Graphs Which Are Uniquely Determined By Neighborhoods
The Zero-Divisor Graphs Which Are Uniquely Determined By Neighborhoods
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DOI：
10.1080/00927870701509156
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发表时间：
2007-11
影响因子：
0.7
通讯作者：
D. Lu;Tongsuo Wu
中科院分区：
文献类型：
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作者：
D. Lu;Tongsuo Wu
一个非空简单连通图G称为唯一确定图，如果G的不同顶点有不同邻域。本文证明了：若R是交换环，则Γ（R）是唯一确定的当且仅当R是布尔环或T（R）是局部环，且对任意x ∈ Z（R），x2 = 0，其中T（R）是R的全商环.确定了任意有限完全图的所有特征为p的相应环，特别是给出了Kn的所有相应环，如果n + 1 = pq，其中p，q是素数.最后，我们证明了如果G是某个布尔环的零因子图，则具有两个以上顶点的图G有唯一对应的零因子半群。
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 Kn if n + 1 = pq 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.