On the essential dot product graph of a commutative ring

Document Type : Original paper

Authors

Mathematics, Science, Aligarh Muslim University, Aligarh, India

Abstract

Let B be a commutative ring with unity 1≠0, 1≤m<∞ be an integer and R=B×B×⋯×B (m times). The total essential dot product graph ETD(R) and the essential zero-divisor dot product graph EZD(R) are undirected graphs with the vertex sets R∗=R∖{(0,0,...0)} and Z(R)∗=Z(R)∖{(0,0,...,0)} respectively. Two distinct vertices w=(w1,w2,...,wm) and z=(z1,z2,...,zm) are adjacent if and only if annB(w⋅z) is an essential ideal of B (where w⋅z=w1z1+w2z2+⋯+wmzm∈B). In this paper, we prove some results on connectedness, diameter and girth of ETD(R) and EZD(R). We classify the ring R such that EZD(R) and ETD(R) are planar, outerplanar, and of genus one.

Keywords

Main Subjects

References

[1] S. Akbari and A. Mohammadian, On the zero-divisor graph of a commutative ring, J. Algebra 274 (2004), no. 2, 847–855.  https://doi.org/10.1016/S0021-8693(03)00435-6
[2] S. Akbari and A. Mohammadian, Zero-divisor graphs of non-commutative rings, J. Algebra 296 (2006), no. 2, 462–479. https://doi.org/10.1016/j.jalgebra.2005.07.007
[3] D.F. Anderson, R. Levy, and J. Shapiro, Zero-divisor graphs, von neumann regular rings, and boolean algebras, J. Pure Appl. Algebra 180 (2003), no. 3, 221–241.  https://doi.org/10.1016/S0022-4049(02)00250-5
[4] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), no. 2, 434–447.  https://doi.org/10.1006/jabr.1998.7840
[5] D.F. Anderson and M. Naseer, Beck’s coloring of a commutative ring, J. Algebra 159 (1993), no. 2, 500–514.  https://doi.org/10.1006/jabr.1993.1171
[6] A. Azadi, M.J. Nikmehr, and B. Soleymanzadeh, Perfect essential graphs, Le Matematiche 74 (2019), no. 2, 279–287.
[7] A. Badawi, On the dot product graph of a commutative ring, Comm. Algebra 43 (2015), no. 1, 43–50.  https://doi.org/10.1080/00927872.2014.897188
[8] J. Battle, F. Harary, and Y. Kodama, Additivity of the genus of a graph, Bull. Amer. Math. Soc. 68 (1962), no. 6, 565–568.
[9] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208–226.  https://doi.org/10.1016/0021-8693(88)90202-5
[10] J.D. LaGrange, Complemented zero-divisor graphs and boolean rings, J. Algebra 315 (2007), no. 2, 600–611.  https://doi.org/10.1016/j.jalgebra.2006.12.030
[11] T.G. Lucas, The diameter of a zero divisor graph, J. Algebra 301 (2006), no. 1, 174–193.  https://doi.org/10.1016/j.jalgebra.2006.01.019
[12] M. Nazim and N. ur Rehman, On the essential annihilating-ideal graph of commutative rings., Ars Math. Contemp. 22 (2022), no. 3, #P3.05.  https://doi.org/10.26493/1855-3974.2645.8fc
[13] M.J. Nikmehr, R. Nikandish, and M. Bakhtyiari, On the essential graph of a commutative ring, J. Algebra Appl. 16 (2017), no. 7, Article ID: 1750132.  https://doi.org/10.1142/S0219498817501328
[14] S. Payrovi, S. Babaei, and E.S. Sevim, On the compressed essential graph of a commutative ring, Bull. Belg. Math. Soc. Simon Stevin 26 (2019), no. 3, 421–429.  https://doi.org/10.36045/bbms/1568685656
[15] K. Selvakumar, M. Subajini, and M.J. Nikmehr, On the genus of essential graph of commutative rings., Australas. J Comb. 74 (2019), no. 1, 74–85.
[16] D.B. West, Introduction to Graph Theory, Prentice hall Upper Saddle River, New Jersey, 1996.
[17] A.T. White, Graphs, Groups and Surfaces, Elsevier, New York, 1985.
Communications in Combinatorics and Optimization
Volume 10, Issue 2
June 2025
Pages 319-334

Files

Share

How to cite

Statistics