The monography examines the problem of constructing a group of automorphisms of a graph. A graph automorphism is a mapping of a set of vertices onto itself that preserves adjacency. The set of such automorphisms forms a vertex group of a graph or simply a graph group. The basis for constructing a group of graph automorphisms is the concept of orbit. The construction of an orbit is closely related to the quantitative assessment of a vertex or edge of a graph, called weight. To determine the weight of an element, graph invariants built on the spectrum of edge cuts and the spectrum of edge cycles are used. The weight of the graph elements allows identifying generating cycles and forming orbits. Examples are given of constructing a group of automorphisms for some types of graphs.