Union-Find algorithm is a classic algorithm used to form the union of disjoined sets and find connected components of a graph from a given set of vertices and edges. The algorithm is often used in data manipulations that involve graphs, trees, hierarchies, and linked networks. A tree data structure can be used to implement the algorithm. A SAS data set is a tuple data structure that is excellen...

in this paper, we introduce the notions of product vague graph, balanced product vague graph, irregularity and total irregularity of any irregular vague graphs and some results are presented. also, density and balanced irregular vague graphs are discussed and some of their properties are established. finally we give an application of vague digraphs.