Super-vertex-antimagic total labelings of disconnected graphs

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