Automorphism group of the one-divisor graph over the semigroup of upper triangular matrices

2020 ◽  
pp. 2150214
Author(s):  
Xinlei Wang ◽  
Dein Wong ◽  
Fenglei Tian
Keyword(s):  
Automorphism Group ◽  
Zero Divisor ◽  
Symmetric Groups ◽  
Directed Edge ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Zero Divisor Graph ◽  
Vertex Set ◽  
The One ◽  
Definition Of

Let [Formula: see text] be a finite field with [Formula: see text] elements, [Formula: see text] a positive integer, [Formula: see text] the semigroup of all [Formula: see text] upper triangular matrices over [Formula: see text] under matrix multiplication, [Formula: see text] the group of all invertible matrices in [Formula: see text], [Formula: see text] the quotient group of [Formula: see text] by its center. The one-divisor graph of [Formula: see text], written as [Formula: see text], is defined to be a directed graph with [Formula: see text] as vertex set, and there is a directed edge from [Formula: see text] to [Formula: see text] if and only if [Formula: see text], i.e. [Formula: see text] and [Formula: see text] are, respectively, a left divisor and a right divisor of a rank one matrix in [Formula: see text]. The definition of [Formula: see text] is motivated by the definition of zero-divisor graph [Formula: see text] of [Formula: see text], which has vertex set of all nonzero zero-divisors in [Formula: see text] and there is a directed edge from a vertex [Formula: see text] to a vertex [Formula: see text] if and only if [Formula: see text], i.e. [Formula: see text]. The automorphism group of zero-divisor graph [Formula: see text] of [Formula: see text] was recently determined by Wang [A note on automorphisms of the zero-divisor graph of upper triangular matrices, Lin. Alg. Appl. 465 (2015) 214–220.]. In this paper, we characterize the automorphism group of one-divisor graph [Formula: see text] of [Formula: see text], proving that [Formula: see text], where [Formula: see text] is the automorphism group of field [Formula: see text], [Formula: see text] is a direct product of some symmetric groups. Besides, an application of automorphisms of [Formula: see text] is given in this paper.

Automorphism group and fixing number of the orthogonality graph based on rank one upper triangular matrices

2020 ◽  
pp. 2250023
Author(s):  
Shikun Ou ◽  
Yanqi Fan ◽  
Fenglei Tian
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Canonical Form ◽  
Undirected Graph ◽  
Induced Subgraph ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Zero Divisors ◽  
Rank One ◽  
Vertex Set

The orthogonality graph [Formula: see text] of a ring [Formula: see text] is the undirected graph with vertex set consisting of all nonzero two-sided zero divisors of [Formula: see text], in which for two vertices [Formula: see text] and [Formula: see text] (needless distinct), [Formula: see text] is an edge if and only if [Formula: see text]. Let [Formula: see text], [Formula: see text] be the set of all [Formula: see text] matrices over a finite field [Formula: see text], and [Formula: see text] the subset of [Formula: see text] consisting of all rank one upper triangular matrices. In this paper, we describe the full automorphism group, and using the technique of generalized equivalent canonical form of matrices, we compute the fixing number of [Formula: see text], the induced subgraph of [Formula: see text] with vertex set [Formula: see text].

scholarly journals Zero-Divisor Graphs with respect to Ideals in Noncommutative Rings

2011 ◽  
Vol 2011 ◽  
pp. 1-7
Author(s):  
Shahabaddin Ebrahimi Atani ◽  
Ahmad Yousefian Darani
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Undirected Graph ◽  
Noncommutative Rings ◽  
Zero Divisor Graph ◽  
Vertex Set ◽  
Definition Of ◽  
The Ideal

Let R be a commutative ring and I an ideal of R. The zero-divisor graph of R with respect to I, denoted ΓI(R), is the undirected graph whose vertex set is {x∈R∖I|xy∈I for some y∈R∖I} with two distinct vertices x and y joined by an edge when xy∈I. In this paper, we extend the definition of the ideal-based zero-divisor graph to noncommutative rings.

A generalized ideal-based zero-divisor graph

2015 ◽  
Vol 14 (06) ◽  
pp. 1550079 ◽  
Author(s):  
M. J. Nikmehr ◽  
S. Khojasteh
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Chromatic Number ◽  
Zero Divisor ◽  
Undirected Graph ◽  
Clique Number ◽  
Proper Ideal ◽  
Graph Structure ◽  
Zero Divisor Graph ◽  
Vertex Set

Let R be a commutative ring with identity, I its proper ideal and M be a unitary R-module. In this paper, we introduce and study a kind of graph structure of an R-module M with respect to proper ideal I, denoted by ΓI(RM) or simply ΓI(M). It is the (undirected) graph with the vertex set M\{0} and two distinct vertices x and y are adjacent if and only if [x : M][y : M] ⊆ I. Clearly, the zero-divisor graph of R is a subgraph of Γ0(R); this is an important result on the definition. We prove that if ann R(M) ⊆ I and H is the subgraph of ΓI(M) induced by the set of all non-isolated vertices, then diam (H) ≤ 3 and gr (ΓI(M)) ∈ {3, 4, ∞}. Also, we prove that if Spec (R) and ω(Γ Nil (R)(M)) are finite, then χ(Γ Nil (R)(M)) ≤ ∣ Spec (R)∣ + ω(Γ Nil (R)(M)). Moreover, for a secondary R-module M and prime ideal P, we determine the chromatic number and the clique number of ΓP(M), where ann R(M) ⊆ P. Among other results, it is proved that for a semisimple R-module M with ann R(M) ⊆ I, ΓI(M) is a forest if and only if ΓI(M) is a union of isolated vertices or a star.

scholarly journals Thek-Zero-Divisor Hypergraph of a Commutative Ring

2007 ◽  
Vol 2007 ◽  
pp. 1-15 ◽  
Author(s):  
Ch. Eslahchi ◽  
A. M. Rahimi
Keyword(s):  
Commutative Ring ◽  
Nilpotent Element ◽  
Zero Divisor ◽  
Clique Number ◽  
Common Point ◽  
Prime Ideals ◽  
Uniform Hypergraph ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set

The concept of the zero-divisor graph of a commutative ring has been studied by many authors, and thek-zero-divisor hypergraph of a commutative ring is a nice abstraction of this concept. Though some of the proofs in this paper are long and detailed, any reader familiar with zero-divisors will be able to read through the exposition and find many of the results quite interesting. LetRbe a commutative ring andkan integer strictly larger than2. Ak-uniform hypergraphHk(R)with the vertex setZ(R,k), the set of allk-zero-divisors inR, is associated toR, where eachk-subset ofZ(R,k)that satisfies thek-zero-divisor condition is an edge inHk(R). It is shown that ifRhas two prime idealsP1andP2with zero their only common point, thenHk(R)is a bipartite (2-colorable) hypergraph with partition setsP1−Z′andP2−Z′, whereZ′is the set of all zero divisors ofRwhich are notk-zero-divisors inR. IfRhas a nonzero nilpotent element, then a lower bound for the clique number ofH3(R)is found. Also, we have shown thatH3(R)is connected with diameter at most 4 wheneverx2≠0for all3-zero-divisorsxofR. Finally, it is shown that for any finite nonlocal ringR, the hypergraphH3(R)is complete if and only ifRis isomorphic toZ2×Z2×Z2.

On the intersection graph of gamma sets in the zero-divisor graph

2015 ◽  
Vol 07 (01) ◽  
pp. 1450067 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
K. Selvakumar
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Intersection Graph ◽  
Graph Theoretic ◽  
Zero Divisor Graph ◽  
Vertex Set

Let R be a commutative ring. The intersection graph of gamma sets in the zero-divisor graph Γ(R) of R is the graph IΓ(R) with vertex set as the collection of all gamma sets of the zero-divisor graph Γ(R) of R and two distinct vertices A and B are adjacent if and only if A ∩ B ≠ ∅. In this paper, we study about various properties of IΓ(R) and investigate the interplay between the graph theoretic properties of IΓ(R) and the ring theoretic properties of R.

scholarly journals Metric and upper dimension of zero divisor graphs associated to commutative rings

2020 ◽  
Vol 12 (1) ◽  
pp. 84-101 ◽  
Author(s):  
S. Pirzada ◽  
M. Aijaz
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Metric Dimension ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set ◽  
Finite Commutative Rings

AbstractLet R be a commutative ring with Z*(R) as the set of non-zero zero divisors. The zero divisor graph of R, denoted by Γ(R), is the graph whose vertex set is Z*(R), where two distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we investigate the metric dimension dim(Γ(R)) and upper dimension dim+(Γ(R)) of zero divisor graphs of commutative rings. For zero divisor graphs Γ(R) associated to finite commutative rings R with unity 1 ≠ 0, we conjecture that dim+(Γ(R)) = dim(Γ(R)), with one exception that {\rm{R}} \cong \Pi {\rm\mathbb{Z}}_2^{\rm{n}}, n ≥ 4. We prove that this conjecture is true for several classes of rings. We also provide combinatorial formulae for computing the metric and upper dimension of zero divisor graphs of certain classes of commutative rings besides giving bounds for the upper dimension of zero divisor graphs of rings.

scholarly journals On Laplacian Eigenvalues of the Zero-Divisor Graph Associated to the Ring of Integers Modulo n

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 482
Author(s):  
Bilal A. Rather ◽  
Shariefuddin Pirzada ◽  
Tariq A. Naikoo ◽  
Yilun Shang
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Simple Graph ◽  
Laplacian Spectrum ◽  
Laplacian Eigenvalues ◽  
Ring Of Integers ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set ◽  
Positive Integers

Given a commutative ring R with identity 1≠0, let the set Z(R) denote the set of zero-divisors and let Z*(R)=Z(R)∖{0} be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by Γ(R), is a simple graph whose vertex set is Z*(R) and each pair of vertices in Z*(R) are adjacent when their product is 0. In this article, we find the structure and Laplacian spectrum of the zero-divisor graphs Γ(Zn) for n=pN1qN2, where p<q are primes and N1,N2 are positive integers.

scholarly journals Distance-Based Topological Polynomials Associated with Zero-Divisor Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Ali Ahmad ◽  
S. C. López
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Wiener Index ◽  
Formula One ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set ◽  
Hosoya Polynomial ◽  
Nonzero Identity

Let R be a commutative ring with nonzero identity and let Z R be its set of zero divisors. The zero-divisor graph of R is the graph Γ R with vertex set V Γ R = Z R ∗ , where Z R ∗ = Z R \ 0 , and edge set E Γ R = x , y :   x ⋅ y = 0 . One of the basic results for these graphs is that Γ R is connected with diameter less than or equal to 3. In this paper, we obtain a few distance-based topological polynomials and indices of zero-divisor graph when the commutative ring is ℤ p 2 q 2 , namely, the Wiener index, the Hosoya polynomial, and the Shultz and the modified Shultz indices and polynomials.

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