In recent years, the notions of discrete homotopy for graphs, such as A-homotopy and \(\times\) -homotopy fundamental groupoid for graphs, have been introduced. This paper develops the notion of a looped fundamental groupoid for graphs to the isotropy group of the vertex v in the fundamental groupoid. We show that it is similar to the classical fundamental group of a topological space and the A-theory of graphs in some sense. Then, we generalize the concept of this theory to the looped fundamental group of weighted reflexive graphs and investigate some properties for the classification of weighted reflexive graphs in various situations. Moreover, we show that it satisfies the Seifert–Van Kampen property, with analogous results similar to the classical theory. Finally, we discuss the conclusion, its applications in the real world, and future works.
