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Let G be a graph of order n and size m. A vertex magic total labeling of G is a one-to-one function f: V(G)∪E(G) → {1, 2, · · ·, n+m} with the property that for each vertex u of G, the sum of the label of u and the labels of all edges incident to u is the same constant, referred to as the magic constant. Such a labeling is even if f [V(G)] = {2, 4, 6, · · ·, 2n}. A graph G is called an even vertex magic if there is an even vertex magic total labeling of G. The primary goal of this paper is to study wheel related graphs with the size greater than the order, which have an even vertex magic total labeling. For every integer n ≥ 3 and t ≥ 1, the t-fold wheel Wn,t is a wheel related graph derived from a wheel Wn by duplicating the t hubs, each adjacent to all rim vertices, and not adjacent to each other. The t-fold wheel Wn,t has a size nt + n that exceeds its order n + t. In this paper, we determine the magic constant of the t-fold wheel Wn,t, the bound of an integer t for the even vertex magic total labeling of the t-fold wheel Wn,t and the conditions for even vertex magic Wn,t, focusing on integers n and t are established. Additionally, we investigate the necessary conditions for the even vertex magic total labeling of the n-fold wheel Wn,n when n is odd and the n-fold wheel Wn,n−2 when n is even. Furthermore, our study explores the characterization of an even vertex magic Wn,t for integer 3 ≤ n ≤ 9. © 2023 the Author(s), licensee AIMS Press. |
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