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

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 C_nʘK₁ where n  ≥ 3. We label both wheel graph and pendant graph using the concept of the anti-magic labelling method of the path graph P_n-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 P_n-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.