Ideal based trace graph of matrices

Year 2020, , 608 - 616, 02.04.2020

Tamizh Chelvam Thirugnanam , M. Sivagami

https://doi.org/10.15672/hujms.478373

Cited By: 5

Abstract

Let $R$ be a commutative ring and $M_n(R)$ be the set of all $n\times n$ matrices over $R$ where $n\geq 2.$ The trace graph of the matrix ring $M_n(R)$ with respect to an ideal $I$ of $R,$ denoted by $\Gamma_{I^t}(M_n(R)),$ is the simple undirected graph with vertex set $M_n(R)\setminus M_n(I)$ and two distinct vertices $A$ and $B$ are adjacent if and only if Tr$(AB) \in I.$ Here Tr$(A)$ represents the trace of the matrix $A.$ In this paper, we exhibit some properties and structure of $\Gamma_{I^t}(M_n(R)).$

Keywords

Trace graph, Matrix ring, Ideal based graph, clique number

References

• [1] F.A.A. Almahdi, K. Louartiti, and M. Tamekkante, The trace graph of the matrix ring over a finite commutative ring, Acta Math. Hungar. 156 (1), 132–144, 2018.
• [2] D.F. Anderson and P.S. Livingston, The zero divisor graph of a commutative ring, J. Algebra 217, 434–447, 1999.
• [3] I. Beck, Coloring of commutative rings, J. Algebra 116, 208–226, 1988.
• [4] G. Chartrand, O.R. Oellermann, Applied and algorithmic graph theory, McGraw-Hill, Inc., New York, 1993.
• [5] I.N. Herstein, Noncommutative rings, Carus Monographs in Mathematics, Math. Assoc. of America, 1968.
• [6] I. Kaplansky, Commutative Rings, University of Chicago Press, Chicago, 1974.
• [7] T.Y. Lam, A first course in noncommutative rings, Graduate Texts in Mathematics, Springer-Verlag, New York 2001.
• [8] H.R. Maimani, M.R. Pournaki, and S. Yassemi, Zero-divisor graph with respect to an ideal, Comm. Algebra 34 (3), 923–929, 2006.
• [9] S.P. Redmond, An ideal-based zero-divisor graph of a commutative ring, Comm. Algebra 31, 4425–4443, 2003.
• [10] M. Sivagami and T. Tamizh Chelvam, On the trace graph of matrices, Acta Math. Hungar. 158 (1), 235–250, 2019, https://doi.org/10.1007/s10474-019-00918-5.
• [11] D.B. West, Introduction to graph theory, Prentice Hall, 2001.