On the Construction of the Reflexive Vertex k-Labeling of Any Graph with Pendant Vertex

I H Agustin,M I Utoyo,Dafik Dafik,Surahmat Surahmat,M Venkatachalam

doi:10.1155/2020/7812812

Abstract

A total k-labeling is a function fe from the edge set to first natural number ke and a function fv from the vertex set to non negative even number up to 2kv, where k=maxke,2kv. A vertex irregular reflexivek-labeling of a simple, undirected, and finite graph G is total k-labeling, if for every two different vertices x and x′ of G, wtx≠wtx′, where wtx=fvx+Σxy∈EGfexy. The minimum k for graph G which has a vertex irregular reflexive k-labeling is called the reflexive vertex strength of the graph G, denoted by rvsG. In this paper, we determined the exact value of the reflexive vertex strength of any graph with pendant vertex which is useful to analyse the reflexive vertex strength on sunlet graph, helm graph, subdivided star graph, and broom graph.

Highlights

We consider a simple and finite graph G (V, E) with vertex set V(G) and edge set E(G)
We have found the lower bound of vertex irregular reflexive strength of any graph G and determined the vertex irregular reflexive strength of graphs with pendant vertex
We have studied the construction of the reflexive vertex k-labeling of any graph with pendant vertex

Summary

Introduction

We consider a simple and finite graph G (V, E) with vertex set V(G) and edge set E(G). K} is called vertex irregular total k-labeling of graph G if the vertex weight wtφ(x) φ(x) + Σxy∈E(G)φ(xy) is distinct for every two different vertices, wtφ(x) ≠ wtφ(y) for x, y ∈ V(G), x ≠ y. A vertex irregular reflexive k-labeling of the graph G is the total k-labeling, for every two different vertices x and x′ of G, wt(x) ≠ wt(x′), where wt(x) fv(x) + Σxy∈E(G)fe(xy).

Results

Conclusion