Mathematics and Statistics Vol. 11(5), pp. 794 - 801
DOI: 10.13189/ms.2023.110505
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Jacobson Graph of Matrix Rings

Siti Humaira ^1,*, Pudji Astuti ², Intan Muchtadi Alamsyah ², Edy Tri Baskoro ^3
1 Doctoral Program of Mathematics, Faculty of Mathematics and Natural Sciences, Bandung Institute of Technology, Indonesia
² Algebra Research Group, Faculty of Mathematics and Natural Sciences, Bandung Institute of Technology, Indonesia
³ Combinatorics Research Group, Faculty of Mathematics and Natural Sciences, Bandung Institute of Technology, Indonesia

ABSTRACT

Some researchers have studied some properties of the Jacobson graph of commutative rings. In this study, we expand these results by examining the Jacobson graph of a non-commutative ring with identity, where we focus on the case of matrix rings. Initially, we update the definition of the Jacobson graph of non-commutative rings as a directed graph. Then we find that the Jacobson graph of the matrix rings case is undirected. We can classify matrices based on rank by viewing the matrix as a linear transformation. The main result is that the order of the matrix rank values will be proportional to the order of the matrix degrees as vertices of the graph. So that one can identify the maximum and minimum degrees in this graph. Sequentially, we describe the graph properties starting from the Jacobson graph of matrices over fields, then expanding to the Jacobson graph of matrices over local commutative rings and the Jacobson graph of matrices over non-local rings. In the end, we also give different results on the Jacobson graph of triangular matrices. The main contribution of this paper is to review the relationship between the aspects of linear algebra in the form of matrix rings and combinatorics in the form of diameter and vertex degree on this graph.

KEYWORDS
Jacobson Graph, Matrix Ring, Diameter, Degree

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Siti Humaira , Pudji Astuti , Intan Muchtadi Alamsyah , Edy Tri Baskoro , "Jacobson Graph of Matrix Rings," Mathematics and Statistics, Vol. 11, No. 5, pp. 794 - 801, 2023. DOI: 10.13189/ms.2023.110505.

(b). APA Format:
Siti Humaira , Pudji Astuti , Intan Muchtadi Alamsyah , Edy Tri Baskoro (2023). Jacobson Graph of Matrix Rings. Mathematics and Statistics, 11(5), 794 - 801. DOI: 10.13189/ms.2023.110505.