Line graph of unit graphs associated with finite commutative rings

Authors

DOI:

https://doi.org/10.22199/issn.0717-6279-4112

Keywords:

Unit graph, Commutative ring, Clique, Chromatic number, Planarity, Hamiltonian

Abstract

For a given graph G, its line graph denoted by L(G) is a graph whose vertex set V (L(G)) = E(G) and {e1, e2} ∈ E(L(G)) if e1 and e2 are incident to a common vertex in G. Let R be a finite commutative ring with nonzero identity and G(R) denotes the unit graph associated with R. In this manuscript, we have studied the line graph L(G(R)) of unit graph G(R)  associated with R. In the course of the investigation, several basic properties, viz., diameter, girth, clique, and chromatic number of L(G(R)) have been determined. Further, we have derived sufficient conditions for L(G(R)) to be Planar and Hamiltonian

Author Biographies

Pranjali, University of Rajasthan.

Dept.  of Mathematics

Amit Kumar, Banasthali Vidyapith.

Dept. of Mathematics and Statistics

Pooja Sharma, Banasthali Vidyapith.

Dept. of Mathematics and Statistics

References

A. Kumar, Pranjali, M. Acharya, and P. Sharma, “Unit graphs having their domination number half their order,” in Recent advancements in graph theory, N. P. Shrimali and N. H. Shah, Eds. Boca Raton, FL: CRC Press, 2020, pp. 205–218.

N. Ashrafi, H. R. Maimani, M. R. Pournaki, and S. Yassemi, “Unit graphs associated with rings”, Communications in algebra, vol. 38, no. 8, pp. 2851–2871, 2010. https://doi.org/10.1080/00927870903095574

R. P. Grimaldi, “Graphs from rings”, Congressus numerantium, vol. 71, pp. 95-103, 1990.

F. Harary, Graph theory. Reading, MA: Addison-Wesley, 1969.

N. Jacobson, Lectures in abstract algebra. New Delhi: East-West Press, 1951.

H. R. Maimani, M. R. Pournaki, and S. Yassemi, “Weakly perfect graphs arising from rings”, Glasgow mathematical journal, vol. 52, no. 3, pp. 417-425, 2010. https://doi.org/10.1017/S0017089510000108

Pranjali, "Graphs associated with commutative rings", Ph.D. Thesis, University of Delhi, Department of Mathematics, 2016.

Pranjali and M. Acharya, “Energy and Wiener index of unit graphs”, Applied mathematics & information sciences, vol. 9, no. 3, pp. 1339-1343, 2015. [On line]. Available: https://bit.ly/3kBeS95

Published

2021-05-10 — Updated on 2021-07-26

How to Cite

[1]
Pranjali, A. Kumar, and P. Sharma, “Line graph of unit graphs associated with finite commutative rings”, Proyecciones (Antofagasta, On line), vol. 40, no. 4, pp. 919-926, Jul. 2021.

Issue

Section

Artículos