论文标题
实现有限交换环的零径都作为阈值图
Realization of zero-divisor graphs of finite commutative rings as threshold graphs
论文作者
论文摘要
令R为统一的有限交换环，令G =（V，e）为简单的图。由γ（r）表示的零划分图是一个简单的图形，其顶点集为r，而当R中的两个顶点x，y \ in r中相邻γ（r），并且仅当$ xy = 0 $。在[10]中，作者研究了图γ（z_n）的拉普拉斯特征值，并为不同的适当分隔D_1，d_2，d_2，\ dots，d_k，n的d_k，他们将集合定义为，a_ {d_i} = n zn：（x，x，x，x，x，d _i}，（x，x，n ever） n。在本文中，我们证明了a_ {d_i}的集合，1 \ leq i \ leq k实际上是组动作的轨道：aut（γ（r））\ times r \ longrightArrow r，其中aut（γ（r））表示γ（γ（R）的自动形态组。我们的主要目的是确定新的阈值图，因为这些图在多个应用领域起着重要作用。对于降低的环R，我们证明γ（r）是且仅当r = f_q或r = f_2 \ times f_q时，γ（r）是连接的阈值图。我们提供一些本地环实现的阈值图。最后，我们表征了所有有限的交换环，其统一性不是阈值。
Let R be a finite commutative ring with unity, and let G = (V, E) be a simple graph. The zero-divisor graph, denoted by Γ(R) is a simple graph with vertex set as R, and two vertices x, y \in R are adjacent in Γ(R) if and only if $xy = 0$. In [10], the authors have studied the Laplacian eigenvalues of the graph Γ(Z_n) and for distinct proper divisors d_1, d_2, \dots, d_k of n, they defined the sets as, A_{d_i} = {x \in Zn : (x, n) = d_i}, where (x, n) denotes the greatest common divisor of x and n. In this paper, we show that the sets A_{d_i}, 1 \leq i \leq k are actually orbits of the group action: Aut(Γ(R)) \times R \longrightarrow R, where Aut(Γ(R)) denotes the automorphism group of Γ(R). Our main objective is to determine new classes of threshold graphs, since these graphs play an important role in several applied areas. For a reduced ring R, we prove that Γ(R) is a connected threshold graph if and only if R = F_q or R = F_2 \times F_q. We provide classes of threshold graphs realized by some classes of local rings. Finally, we characterize all finite commutative rings with unity of which zero-divisor graphs are not threshold.
