On the Dot Product Graph of a Commutative Ring
On the Dot Product Graph of a Commutative Ring
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DOI：
10.1080/00927872.2014.897188
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发表时间：
2015-01
影响因子：
0.7
通讯作者：
Ayman Badawi
中科院分区：
文献类型：
--
作者：
Ayman Badawi
设 A 为非零恒等式交换环，1 ≤ n < ∞ 为整数，且 R = A × A × … ×A（n 次）。 R 的总点积图是（无向）图 TD(R)，其顶点 R* = R∖{(0, 0,…, 0)}，并且两个不同的顶点 x 和 y 相邻当且仅当 x·y = 0 ∈ A（其中 x·y 表示 x 和 y 的正常点积）。令Z(R)表示R的所有零因数的集合。则R的零因数点积图是TD(R)的导出子图ZD(R)，顶点为Z(R)* = Z(R)∖{(0, 0,…, 0)}。由此可见，经典零因数图 Γ(R) 的每条边（路径）都是 ZD(R) 的一条边（路径）。我们观察到，如果 n = 1，则 TD(R) 是一个不连通图，而 ZD(R) 与 Beck-Anderson-Livingston 意义上的著名的 R 零除数图相同，因此它是连通的。在本文中，我们研究图 TD(R) 和 ZD(R)。对于交换环 A 且 n ≥ 3，我们证明 TD(R) (ZD(R)) 与直径 2（最多 3）和周长 3 相连。除此之外，对于 n ≥ 2，我们证明 ZD(R) 与 R 的零除数图相同当且仅当 n = 2 并且 A 是积分域或 R 与 ℤ2 × ℤ2 × ℤ2 环同构。
Let A be a commutative ring with nonzero identity, 1 ≤ n < ∞ be an integer, and R = A × A × … ×A (n times). The total dot product graph of R is the (undirected) graph TD(R) with vertices R* = R∖{(0, 0,…, 0)}, and two distinct vertices x and y are adjacent if and only if x·y = 0 ∈ A (where x·y denote the normal dot product of x and y). Let Z(R) denote the set of all zero-divisors of R. Then the zero-divisor dot product graph of R is the induced subgraph ZD(R) of TD(R) with vertices Z(R)* = Z(R)∖{(0, 0,…, 0)}. It follows that each edge (path) of the classical zero-divisor graph Γ(R) is an edge (path) of ZD(R). We observe that if n = 1, then TD(R) is a disconnected graph and ZD(R) is identical to the well-known zero-divisor graph of R in the sense of Beck–Anderson–Livingston, and hence it is connected. In this paper, we study both graphs TD(R) and ZD(R). For a commutative ring A and n ≥ 3, we show that TD(R) (ZD(R)) is connected with diameter two (at most three) and with girth three. Among other things, for n ≥ 2, we show that ZD(R) is identical to the zero-divisor graph of R if and only if either n = 2 and A is an integral domain or R is ring-isomorphic to ℤ2 × ℤ2 × ℤ2.