Super-vertex-antimagic total labelings of disconnected graphs

Let G=(V,E)G=(V,E) be a finite, simple and non-empty (p,q)(p,q)-graph of order pp and size qq. An (a,d)(a,d)-vertex-antimagic total labeling is a bijection ff from V(G)∪E(G)V(G)∪E(G) onto the set of consecutive integers 1,2,…,p+q1,2,…,p+q, such that the vertex-weights form an arithmetic progression with the initial term aa and the common difference dd, where the vertex-weight of xx is the sum of values f(xy)f(xy) assigned to all edges xyxy incident to vertex xx together with the value assigned to xx itself, i.e. f(x)f(x). Such a labeling is called super if the smallest possible labels appear on the vertices.In this paper, we will study the properties of such labelings and examine their existence for disconnected graphs.