Line graph of unit graphs associated with finite commutative rings

Pranjali Pranjali,Amit Kumar,Pooja Sharma

doi:10.22199/issn.0717-6279-4112

Pranjali Pranjali, Amit Kumar + Show 1 more

Open Access

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

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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

Highlights

Let n be a positive integer and Zn be the ring of integers modulo n
In 1990, Grimaldi [3] introduced the notion of unit graph denoted as G(Zn) based on the elements of Zn and two distinct vertices x and y are adjacent if and only if x + y is a unit of Zn
This investigation was further continued by Ashrafi et al [2], where author’s were interested to generalize the unit graph G(Zn) to G(R) for an arbitrary associative ring R

Summary

Introduction

Let n be a positive integer and Zn be the ring of integers modulo n. Let R be a finite commutative ring. We determine several basic properties of line graph L(G(R)) of unit graphs G(R) associated with finite commutative rings R.

Results

Conclusion