A simple graph \( G = (V(G), E(G)) \) admits an \( H \)-covering if every edge in \( E(G) \) belongs to a subgraph of \( G \) that is isomorphic to \( H \). An \((a,d)\)-\( H \)-\({antimagic\; total \;labeling}\) of \( G \) is a bijective function \( \xi : V(G) \cup E(G) \to \{1,2,\dots,|V(G)| + |E(G)|\} \) such that for all subgraphs \( H’ \) isomorphic to \( H \), the \( H \)-weights \( w(H’) = \sum_{v \in V(H’)} \xi(v) + \sum_{e \in E(H’)} \xi(e) \) constitute an arithmetic progression \( a, a+d, a+2d, \dots, a+(t-1)d \), where \( a \) and \( d \) are positive integers and \( t \) is the number of subgraphs of \( G \) isomorphic to \( H \). Additionally, the labeling \( \xi \) is called a \({super}\) \((a, d)\)-\( H \)-\({antimagic\; total\; labeling}\) if \( \xi(V(G)) = \{1, 2, \dots, |V(G)|\} \).
In this paper, we introduce the notion of \((a, d)\)-\( H \)-\({antimagic\; total\; labeling}\) and study some basic properties of such labeling. We provide an example of a family of graphs obtaining the labelings, that is providing \((a, d)\)-cycle-antimagic labelings of fans.