Studies in algebraic graph theory showcase the interplay between group theory and graph theory by defining graphs on groups, investigating their properties, and also by analysing the automorphism groups that emerge from the graphs. In this article, we introduce the idea of constructing an algebraic derived graph; that is, constructing a graph based on the algebraic properties of a graph, by introducing the invariant intersection graph of a graph, constructed based on the automorphism group of a graph. Here, we introduce the graph construction and initiate an investigation on the structure of the invariant intersection graph with respect to the graph and its automorphism group.