Eulerian of the Zero Divisor graph $Γ[\mathbb {Z}_n]$

Kaplansky's zero divisor and unit conjectures on elements with supports of size $3$

Kaplansky's zero divisor conjecture (unit conjecture, respectively) states that for a torsion-free group $G$ and a field $\mathbb{F}$, the group ring $\mathbb{F}[G]$ has no zero divisors (has no unit with support of size greater than $1$). In this paper, we study possible zero divisors and units in $\mathbb{F}[G]$ whose supports have size $3$. For any field $\mathbb{F}$ and all torsion-free groups $G$, we prove that if $αβ=0$ for some non-zero $α, β\in \mathbb{F}[G]$ such that $|supp(α)|=3$, then $|supp(β)|\geq 10$. If $\mathbb{F}=\mathbb{F}_2$ is the field with 2 elements, the latter result c…

关于Cayley-Dickson代数中的双交替零因子

On doubly alternative zero divisors in Cayley-Dickson algebras

本文研究特征非2的域上Cayley-Dickson代数的零因子，明确双交替零因子的性质与显式形式，刻画各向异性范数情形下零因子图的结构，得到C等价准则及零化子维数的可能取值。

整数模n环的零因子图中的割边和中心顶点

Cut edges and Central vertices of zero divisor graph of the ring of integers modulo n

本文研究了整数模n环的零因子图中的割边和中心顶点，确定了这些图的结构特性。

Zero divisor and unit elements with support of size 4 in group algebras of torsion free groups

Kaplansky Zero Divisor Conjecture states that if $G $ is a torsion free group and $ \mathbb{F} $ is a field, then the group ring $\mathbb{F}[G]$ contains no zero divisor and Kaplansky Unit Conjecture states that if $G $ is a torsion free group and $ \mathbb{F} $ is a field, then $\mathbb{F}[G]$ contains no non-trivial units. The support of an element $ α= \sum_{x\in G}α_xx$ in $\mathbb{F}[G] $, denoted by $supp(α)$, is the set $ \{x \in G|α_x\neq 0\} $. In this paper we study possible zero divisors and units with supports of size $ 4 $ in $\mathbb{F}[G]$. We prove that if $ α, β$ are non-zero …

Eigenvalues of zero-divisor graphs of finite commutative rings

We investigate eigenvalues of the zero-divisor graph $Γ(R)$ of finite commutative rings $R$ and study the interplay between these eigenvalues, the ring-theoretic properties of $R$ and the graph-theoretic properties of $Γ(R)$. The graph $Γ(R)$ is defined as the graph with vertex set consisting of all non-zero zero-divisors of $R$ and adjacent vertices $x,y$ whenever $xy = 0$. We provide formulas for the nullity of $Γ(R)$, i.e. the multiplicity of the eigenvalue 0 of $Γ(R)$. Moreover, we precisely determine the spectra of $Γ(\mathbb Z_p \times \mathbb Z_p \times \mathbb Z_p)$ and $Γ(\mathbb Z_p …

The characteristic equation and Wiener index of a compressed zero divisor graph

The Zero divisor Graph of a commutative ring $R$, denoted by $Γ[R]$, is a graph whose vertices are non-zero zero divisors of R and two vertices are adjacent if their product is zero. The compressed zero divisor graph $Γ_E[R]$ is the (undirected) graph whose vertices are the equivalence classes such that distinct vertices [r] and [s] are adjacent if and only if rs = 0. In this paper we derive the characteristic polynomial and Wiener index of the Compressed zero divisor graph $Γ_{E}[\mathbb{Z}_m]$ where $m=p^n$ with prime $p$.

关于 $\mathbb Z_{2^n}$ 上 Hamilton 四元数的非零因子图

On the non-zero divisor graph of the Hamilton quaternions over $\mathbb Z_{2^n}$

本文研究环 $\mathbb H(\mathbb Z_{2^n})$ 的非零因子图，通过分析元素结构探讨图的连通性、邻接性、可遍历性和平面性，并给出团数和色数的界。

On the cozero-divisor graphs assosciated to rings

Let $R$ be a ring with unity. The cozero-divisor graph of a ring $R$, denoted by $Γ'(R)$, is an undirected simple graph whose vertices are the set of all non-zero and non-unit elements of $R$, and two distinct vertices $x$ and $y$ are adjacent if and only if $x \notin Ry$ and $y \notin Rx$. In this paper, first we study the Laplacian spectrum of $Γ'(\mathbb{Z}_n)$. We show that the graph $Γ'(\mathbb{Z}_{pq})$ is Laplacian integral. Further, we obtain the Laplacian spectrum of $Γ'(\mathbb{Z}_n)$ for $n = p^{n_1}q^{n_2}$, where $n_1, n_2 \in \mathbb{N}$ and $p, q$ are distinct primes. In order t…

On Common Divisors of Fox Derivatives with towards to Zero Divisors of Group Rings

Using Composition--Diamond Lemma we construct presentations of groups $G = \langle x_1,\ldots,x_n \, | \, r_1,\ldots, r_m \rangle$ with the following property; for a fixed $1 \le i \le n$, and for all $1 \le j \le m$, Fox derivatives $\partial r_j / \partial x_i$ have common divisor. It follows that if $π_2(K) \ne 0$, where $K$ is the standard $2$-complex associated with $G$ then the group ring $\mathbb{Z}[G]$ has nontrivial zero divisors.

The Variance and Correlations of the Divisor Function in $\mathbb{F}_q [T]$, and Hankel Matrices

We prove an exact formula for the variance of the divisor function over short intervals in $\mathcal{A} := \mathbb{F}_q [T]$, where $q$ is a prime power. A slight adaption of the proof allows us to obtain an exact formula for correlations of the form $d(A) d(A+B)$, where we average both $A$ and $B$ over certain intervals in $\mathcal{A}$. We also consider correlations of the form $d(KQ+N) d (N)$, where $Q$ is prime and $K$ and $N$ are averaged over certain intervals. If $\mathrm{deg } K < \mathrm{deg } Q -1$, then these correlations appear in the off-diagonal terms for the fourth moment of Dir…

交换环的弱零因子图的Sombor指标与特征值

Sombor index and eigenvalues of weakly zero-divisor graph of commutative rings

本文研究交换环的弱零因子图，确定了整数模环的Sombor指标，并给出其Sombor谱与能量的界。

On realizing zero-divisor graphs of po-semirings

In this paper, we determine bipartite graphs and complete graphs with horns, which are realizable as zero-divisor graphs of po-semirings. As applications, we classify commutative rings $R$ whose annihilating-ideal graph $\mathbb {AG}(R)$ are either bipartite graphs or complete graphs with horns.

关于交换环零因子图特征值的一个注记

A note on eigenvalues of zero divisor graphs associated with commutative rings

本文纠正了交换环零因子图特征值、能量及Zagreb指标已有结果的错误，并证明特定环的零因子图满足相应能量性质和Hansen-Vukičević猜想。