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.

翻译：在本专著中，我们提出将图的边割谱与边环谱概念作为解决图同构问题的基本数学结构。边割由一条边及其关联顶点定义。与迭代边图的生成方式不同，我们考虑由边割生成并通过递推关系确定的原始图迭代链。边环由图的等距环集合定义。本专著研究了构建图G的边割谱Ws与边环谱Tc的方法。研究表明，谱的构建基于图的关联矩阵。证明了图结构的构建与顶点及边编号的独立性。论证了边割谱与边环谱对于判定图结构同构的必要性与充分性。探讨了图内部结构与Whitney定理之间的关系。