Super \( a, d \)-vertex-antimagic Total Labelings

Let \( G = (V, E) \) be a graph with \( v \) vertices and \( e \) edges. A \((a, d)\)-\text{vertex-antimagic total labeling} is a bijection \( \alpha \) from \( V(G) \cup E(G) \) to the set of consecutive integers \( 1, 2, \ldots, v+e \), such that the weights of the vertices form an arithmetic progression with the initial term \( a \) and the common difference \( d \). If \( \alpha(V(G)) = \{1, 2, \ldots, v\} \), then we call the labeling a \text{super \((a, d)\)-vertex antimagic total}. We study basic properties of such labelings and show how to construct such labelings for some families of graphs, such as paths, cycles, and generalized Petersen graphs. We also show that such labeling does not exist for certain families of graphs, such as cycles with at least one tail, trees with an even number of vertices, and all stars.