Some new results on Jacobson graphs

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 J ( R ) m × n , 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.

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The matrix Jacobson graph of fields

Humaira

Astuti

Muchtadi-Alamsyah

et al. 2020

J. Phys.: Conf. Ser.

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Jacobson graph over Z _n

Aditya

Muchtadi-Alamsyah

2021

J. Phys.: Conf. Ser.

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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.

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Some new results on Jacobson graphs

Cited by 2 publications

References 4 publications

The matrix Jacobson graph of fields

The matrix Jacobson graph of fields

Jacobson graph over Z _n

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Some new results on Jacobson graphs

Cited by 2 publications

References 4 publications

The matrix Jacobson graph of fields

The matrix Jacobson graph of fields

Jacobson graph over Z n

Contact Info

Product

Resources

About

Jacobson graph over Z _n