The even vertex magic total labelings of $ t $-fold wheels

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) \cup E(G) \rightarrow \{1, 2, \cdots, 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, \cdots, 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 \geq 3 $ and $ t \geq 1 $, the $ t $-fold wheel $ W_{n, t} $ is a wheel related graph derived from a wheel $ W_n $ by duplicating the $ t $ hubs, each adjacent to all rim vertices, and not adjacent to each other. The $ t $-fold wheel $ W_{n, 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 $ W_{n, t} $, the bound of an integer $ t $ for the even vertex magic total labeling of the $ t $-fold wheel $ W_{n, t} $ and the conditions for even vertex magic $ W_{n, 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 $ W_{n, n} $ when $ n $ is odd and the $ n $-fold wheel $ W_{n, n-2} $ when $ n $ is even. Furthermore, our study explores the characterization of an even vertex magic $ W_{n, t} $ for integer $ 3 \leq n \leq 9 $.