An alternative approach for the anti-magic labelling of a wheel graph and a pendant graph

K M P G S C Kapuhennayake Kapuhennayake,A A I Perera

doi:10.4038/jsc.v12i2.35

Abstract

The Anti-magic labelling of a graph G with m edges and vertices, is a bijection from the set of edges to the set of integers {1, …, m} such that all ‘n’ vertex summations are pairwise distinct. The vertex summation is the summation of the labels assigned to edges incident to a vertex. There is a conjecture that all simple connected graphs except K2 are anti-magic. In our research, we found an alternative anti-magic labelling method for a wheel graph and a pendant graph. Wheel graph is a graph that contains a cycle of length n - 1 and for which every graph vertex in the cycle is connected to one other graph vertex known as the “hub”. The edges of a wheel, which connect to the hub are called “spokes”. Pendant graph is a corona of the form CnʘK1 where n ≥ 3. We label both wheel graph and pendant graph using the concept of the anti-magic labelling method of the path graph Pn-1. For wheel graph, we removed the middle vertex of the wheel graph and created a path graph using the vertices in the outer cycle of the wheel graph. Then the spokes of the wheel graph are represented by adding one edge to each vertex. For Pendant graph, we created a path graph using the cycle of the pendant graph and connect the pendant vertices to every vertex of the path graph. In both cases, we label all the edges using the concept of the anti-magic labelling of path graph Pn-1. Finally, we calculated the vertex sum for each vertex and proved that every vertex sums are distinct and in the wheel graph, middle vertex takes the highest value.

Highlights

Anti-Magic labelling comes from its connection to magic labelling and magic squares
We removed the middle vertex of the wheel graph and created a path graph using the vertices in the outer cycle of the wheel graph
We introduce an alternative method for anti-magic labelling of wheel graph and pendant graph using the anti-magic labelling of path graph (Chang, Chen, & Li, 2021)

Summary

INTRODUCTION

Anti-Magic labelling comes from its connection to magic labelling and magic squares. A magic square is a square array of numbers consisting of the distinct positive integers. Arranged such that the sum of numbers in any horizontal, vertical and main diagonal lines are always the same number. That is arranging numbers in a way such that the sums of numbers in the horizontal, vertical and main diagonal lines are distinct. The concept of anti-magic labelling was introduced by Hartsfield and Ringel in 1989. (Hartsfield & Ringel, Pearls in graph theory, 1990). They defined it as follows: An Anti-. Magic labelling of a graph with edges and vertices, is a bijection from the set of edges to the set of integers such that all ‘ ’ vertex summations are pairwise distinct. The vertex sum is the summation of labels of all edges incident with that vertex They conjectured that all simple connected graphs except are anti-magic We introduce an alternative method for anti-magic labelling of wheel graph and pendant graph using the anti-magic labelling of path graph (Chang, Chen, & Li, 2021)

MATERIAL AND METHODS

RESULTS AND DISCUSSION

CONCLUSION