On the genus of dot product graph of a commutative ring

K. Selvakumar, V. Ramanathan () and C. Selvaraj
Additional contact information
K. Selvakumar: Manonmaniam Sundaranar University
V. Ramanathan: Manonmaniam Sundaranar University
C. Selvaraj: Periyar University

Indian Journal of Pure and Applied Mathematics, 2023, vol. 54, issue 2, 558-567

Abstract: Abstract In this paper, we study the topological graph theoretic properties such as planarity, toroidality and bi-toroidality of the total dot product graph of a commutative ring. In particular, we characterize an isomorphism class of commutative rings R for which TD(R) has genus one or two. This leads to the characterization of all commutative rings whose ZD(R) has genus one or two. It is shown that for any commutative ring R, TD(R) is a bi-toroidal graph if and only if R is ring isomorphic to $$\frac{{\mathbb {Z}}_2\left[ x\right] }{\left\langle x^2+x+1\right\rangle }\times \frac{{\mathbb {Z}}_2\left[ x\right] }{\left\langle x^2+x+1\right\rangle }$$ Z 2 x x 2 + x + 1 × Z 2 x x 2 + x + 1 and $${\mathbb {Z}}_5\times {\mathbb {Z}}_5.$$ Z 5 × Z 5 .

Keywords: Total dot product graph; Zero-divisor dot product graph; Planar graph; Genus; 13A15; 13M05; 05C75; 05C25 (search for similar items in EconPapers)
Date: 2023
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s13226-022-00275-0 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:indpam:v:54:y:2023:i:2:d:10.1007_s13226-022-00275-0

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/13226

DOI: 10.1007/s13226-022-00275-0

Access Statistics for this article

Indian Journal of Pure and Applied Mathematics is currently edited by Nidhi Chandhoke

More articles in Indian Journal of Pure and Applied Mathematics from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().