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 $f$: $V(G)\cup E(G)\to \{1,2,\cdots ,n+m\}$ 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 $n\ge 3$ and $t\ge 1$, 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 $3\le n\le 9$.