Jacobson Graph Research Articles

Jacobson Graph of Matrix Rings

Jacobson Graph of Matrix Rings

The matrix Jacobson graph of finite commutative rings

The notion of the matrix Jacobson graph was introduced in 2019. Let R be a commutative ring and J ( R ) be the Jacobson radical of ring R . The matrix Jacobson graph of ring R size m × n , denoted 𝔍( R ) m × n , is defined as a graph where the vertex set is R m × n ∖ J ( R ) m × n such that two distinct vertices A , B are adjacent if and only if 1 − det( A t B ) is not a unit in ring R . Here we obtain some graph theoretical properties of 𝔍( R ) m × n including its connectivity, planarity and perfectness.

Jacobson graph over Z n

Let n ≥ 2. A Jacobson graph over ℤ n is a graph with vertex set all elements in ℤ n except elements in its Jacobson radical, and x and y are adjacent if and only if 1 − xy is not relative prime to n. In this paper, we give a characterization of Jacobson graph over ℤ n as a foundation for Jacobson graph over a finite commutative ring.

The matrix Jacobson graph of fields

The notion of Jacobson graph and n-array Jacobson graph of a commutative ring were introduced in 2012 and 2018, respectively, by Azimi et al and Ghayour et al. In this article we generalize them to matrix Jacobson graph. Let R be a commutative ring. The matrix Jacobson graph of a ring R, denoted , is defined as a graph with vertex set is the set of matrix of ring without the matrix of its Jacobson such that two distinct vertices A, B are adjacent if and only if 1 − det(AtB) is not a unit of ring. In this article we study the matrix Jacobson graph where the underlying ring R is a finite field. Since any matrix of size m × n over a field F can be considered as a linear mapping from linear space Fm to Fn , we employ the structure of linear mappings on finite dimensional vector spaces to derive some properties of square and non square matrix Jacobson graph of fields, including their diameters.

Jacobson graph construction of ring , for n>1

A group is called a ring if the group is a commutative under addition operation and satisfy the distributive and assosiative properties under multiplication operation. Suppose R is a commutative ring with non-zero identity, U is the unit of R, and J(R) is a Jacobson radical. Jacobson graph of a ring R denoted by ℑ R is a graph with a vertex set is R\\J(R) dan edge set is {(a, b)| 1 − ab ∉ U}. The purpose of this research is to construct a Jacobson graph of ring Z 3 n with n > 1. The results show that Jacobson graph of ring Z 3 n is a disconected graph with two components.

The genus two class of graphs arising from rings

Given a commutative ring [Formula: see text] with identity [Formula: see text], its Jacobson graph [Formula: see text] is defined to be the graph in which the vertex set is [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Here [Formula: see text] denotes the Jacobson radical of [Formula: see text] and [Formula: see text] is the set of unit elements in [Formula: see text]. This paper investigates the genus properties of Jacobson graph. In particular, we determine all isomorphism classes of commutative rings whose Jacobson graph has genus two.

Chordality of graphs associated to commutative rings

We investigate when different graphs associated to commutative rings are chordal. In particular, we characterize commutative rings $R$ with each of the following conditions: the total graph of $R$ is chordal; the total dot product or the zero-divisor dot product graph of $R$ is chordal; the comaximal graph of $R$ is chordal; $R$ is semilocal; and the unit graph or the Jacobson graph of $R$ is chordal. Moreover, we state an equivalent condition for the chordality of the zero-divisor graph of an indecomposable ring and classify decomposable rings that have a chordal zero-divisor graph.

When is the Jacobson graph projective?

Let [Formula: see text] be a commutative ring with nonzero identity and [Formula: see text] be the Jacobson radical of [Formula: see text]. The Jacobson graph of [Formula: see text], denoted by [Formula: see text], is a graph with vertex-set [Formula: see text], such that two distinct vertices [Formula: see text] and [Formula: see text] in [Formula: see text] are adjacent if and only if [Formula: see text] is not a unit of [Formula: see text]. The goal in this paper is to list every finite commutative ring [Formula: see text] with nonzero identity (up to isomorphism) such that the graph [Formula: see text] is projective.

Some new results on Jacobson graphs

Let R be a commutative ring with non-zero identity, and J(R) be the Jacobson ideal of R. The Jacobson graph of R is a graph with vertex set $$R {\setminus } J(R),$$ and two distinct vertices x and y are adjacent if and only if $$1-xy $$ is not a unit element. In this paper we characterize the finite Jacobson graphs which are chordal graphs, cographs, line graphs, or interval graphs. Among other results, we find the degree set of finite Jacobson graphs, and the number of vertices with specific degree.

Some results on the Jacobson graph of a commutative ring

Let R be a commutative ring with non-zero identity and J(R) be Jacobson ideal of R. The Jacobson graph of R is the graph whose vertices are \(R{\setminus } J(R)\), and two different vertices x and y are adjacent if \(1-xy\notin U(R)\), where U(R) is the set of units of R. We investigate diameter of \(\mathfrak {J}_R\) and seek relation between it and diameter of Jacobson graphs under extension to polynomial and power series rings. Also, vertex and edge connectivity of finite Jacobson graphs are obtained. Finally, we show that all finite Jacobson graphs have a matching that misses at most one vertex and offer one 1-factor decomposition of a regular induced subgraph.

Classification of rings with toroidal Jacobson graph

Let R be a commutative ring with nonzero identity and J(R) the Jacobson radical of R. The Jacobson graph of R, denoted by JR, is defined as the graph with vertex set RJ(R) such that two distinct vertices x and y are adjacent if and only if 1 − xy is not a unit of R. The genus of a simple graph G is the smallest nonnegative integer n such that G can be embedded into an orientable surface Sn. In this paper, we investigate the genus number of the compact Riemann surface in which JR can be embedded and explicitly determine all finite commutative rings R (up to isomorphism) such that JR is toroidal.

The nonorientable genus of some Jacobson graphs and classification of the projective ones

The nonorientable genus of some Jacobson graphs and classification of the projective ones

Classification of the Toroidal Jacobson Graphs

Let R be a finite commutative ring with nonzero identity and denote its Jacobson radical by J(R). The Jacobson graph of R is the graph in which the vertex set is $$R\setminus J(R)$$ , and two distinct vertices x and y are adjacent if and only if $$1-xy$$ is not a unit in R. In this paper, up to isomorphism, we classify the rings R whose Jacobson graphs are toroidal.

Classification of finite commutative rings with planar, toroidal, and projective line graphs associated with Jacobson graphs

Let R be a commutative ring with nonzero identity and J(R) be the Jacobson radical of R. The Jacobson graph of R, denoted by JR, is a graph with vertex-set RJ(R), such that two distinct vertices a and b in RJ(R) are adjacent if and only if 1 − ab is not a unit of R. Also, the line graph of the Jacobson graph is denoted by L(JR). In this paper, we characterize all finite commutative rings R such that the graphs L(JR) are planar, toroidal or projective.

The proof of a conjecture in Jacobson graph of a commutative ring

Let R be a commutative ring with nonzero identity. The Jacobson graph of R denoted by 𝔍R is a graph with the vertex set R\J(R), and two distinct vertices x, y ∈ V(𝔍R) are adjacent if and only if 1 - xy ∉ U(R), where U(R) is the set of all unit elements of R. Let ω(𝔍R) denote the clique number of 𝔍R. It was conjectured that if [Formula: see text] is a commutative finite ring and (Ri, 𝔪i) is a local ring, for i = 1, …, n, then [Formula: see text], where Fi = Ri/𝔪i, for i = 1, …, n. In this paper, we prove that if R is a commutative ring (not necessarily finite) and R is not a field, then ω(𝔍R) = max 𝔪∈ Max (R) |𝔪| and using this we show that the aforementioned conjecture holds.

Isomorphisms between Jacobson graphs

Let \(R\) be a commutative ring with a non-zero identity and \(\mathfrak {J}_R\) be its Jacobson graph. We show that if \(R\) and \(R'\) are finite commutative rings, then \(\mathfrak {J}_R\cong \mathfrak {J}_{R'}\) if and only if \(|J(R)|=|J(R')|\) and \(R/J(R)\cong R'/J(R')\). Also, for a Jacobson graph \(\mathfrak {J}_R\), we obtain the structure of group \(\mathrm {Aut}(\mathfrak {J}_R)\) of all automorphisms of \(\mathfrak {J}_R\) and prove that under some conditions two semi-simple rings \(R\) and \(R'\) are isomorphic if and only if \(\mathrm {Aut}(\mathfrak {J}_R)\cong \mathrm {Aut}(\mathfrak {J}_{R'})\).

THE JACOBSON GRAPH OF COMMUTATIVE RINGS

Let R be a commutative ring with nonzero identity. Then the Jacobson graph of R, denoted by 𝔍R, is defined as a graph with vertex set R\J(R) such that two distinct vertices x and y are adjacent if and only if 1 - xy is not a unit of R. We obtain some graph theoretical properties of 𝔍R including its connectivity, planarity and perfectness and we compute some of its numerical invariants, namely diameter, girth, dominating number, independence number and vertex chromatic number and give an estimate for its edge chromatic number.