  In this monography, it is proposed to consider the concepts of spectra of
edge cuts and edge cycles of a graph as a basic mathematical structure for
solving the problem of graph isomorphism. An edge cut is defined by an edge and
the vertices incident to it. In contrast to the generation of iterated edge
graphs, we consider an iterated chain of qualicuts of the original graph,
generated by edge cuts and determined by a recurrence relation. An edge cycle
is defined by the set of isometric cycles of a graph. The monography examines
the issues of constructing the spectrum of edge cuts Ws and the spectrum of
edge cycles Tc of a graph G. It is shown that the formation of spectra is based
on the incidence matrix of the graph. The independence of the construction of
the graph structure from the numbering of vertices and edges is shown. The
necessity and sufficiency of the spectra of edge cuts and the spectrum of edge
cycles for determining the isomorphism of graph structures is shown. The
relation between the internal structures of the graph and Whitney's theorem is
considered.
