Super Edge-Graceful Labelings of Total Stars and Total Cycles

Let \( [n]^* \) denote the set of integers \(\{-\frac{n-1}{2}, \ldots, \frac{n-1}{2}\}\) if \(n\) is odd, and \(\{-\frac{n}{2}, \ldots, \frac{n}{2}\} \setminus \{0\}\) if \(n\) is even. A super edge-graceful labeling \(f\) of a graph \(G\) of order \(p\) and size \(q\) is a bijection \(f : E(G) \to [q]^*\), such that the induced vertex labeling \(f^*\) given by \(f^*(u) = \sum_{uv \in E(G)} f(uv)\) is a bijection \(f^* : V(G) \to [p]^*\). A graph is super edge-graceful if it has a super edge-graceful labeling. We prove that total stars and total cycles are super edge-graceful.