Petrov Andrey Evgenievich, Doctor of Technical Sciences, Professor of the Department of Computer-Aided Design and Engineering, Federal State Budgetary Educational Institution of Higher Education “National University of Science and Technology “MISiS” (NUST MISIS)
Abstract
Duality for nonplanar networks is a fundamental problem that remains unresolved. Kuratowski [5] showed that there is no graph or network dual to a nonplanar network. He demonstrated that there are only two such graphs that cannot be represented on a plane without self-intersections: the bipartite graph on six vertices with nine edges and the complete graph on five vertices with ten edges. The structure of networks dual to nonplanar networks cannot be represented, but it turns out that they correspond to transformation matrices that can be used to calculate responses to influences. Thus, there are no networks, but flows in them do exist, although they have strange properties. Since the dual network can (must) be located in a space dual to the observed space where the given network resides, nonplanar networks can be the contact point between these spaces.
A dual network corresponds to a path transformation matrix that is orthogonal to the transformation matrix of the original network. The strange thing about nonplanarity is that, by connecting individual branches (nine or ten) into increasingly complex networks, a dual network is always obtained. This can be constructed on a plane as a dual graph, or using a transformation matrix orthogonal to the given network. When connecting branches and reaching a nonplanar network, the dual network “disappears,” but with further connections, it reappears.
One might expect that since there is no dual network for a nonplanar network, there would also be no transformation matrix for its paths when going from the simplest network of individual branches to a connected network. Or perhaps this matrix has some singularities. It turns out that the transformation matrix of a nonplanar network has an inverse matrix, meaning there are no singularities. Transposing this inverse matrix yields an orthogonal matrix, which should correspond to the transformation matrix of the dual network.
Theoretically, it is impossible to construct a network dual to a nonplanar network using this matrix. However, when attempting to construct it, it turns out that the “dual” network of a bipartite graph on six vertices is similar to the complete graph on five vertices, although not completely – one branch is missing. When attempting to construct a “dual” network of a complete graph on five vertices, the resulting network is similar to the bipartite graph on six vertices, although not completely – one branch appears to be superfluous.
Thus, the transformation matrix of a nonplanar network exists, indicating that a corresponding network must also exist. Using the transformation matrix of a network dual to a nonplanar network, one can calculate the network’s solution matrix, represented by the metric tensor of its structure, although its form cannot be explicitly represented. This article discusses the results of research into the structure and processes in networks dual to nonplanar networks that cannot be represented. The practical interest of this problem lies in the fact that a bipartite graph on six vertices can be interpreted as three product-producing industries connected by supplies, i.e., this graph is at the core of economics and its model. The money supply network is dual to the product production network. A nonplanar graph in the product flow network can generate inconsistencies and contradictions in the money supply network.
KEYWORDS: nonplanar graph, dual network, transformation matrix, duality invariant, energy flow conservation law.
Download article NONPLANAR GRAPHS, DUAL NETWORKS, THE LAW OF CONSERVATION OF ENERGY FLOW: A FUNDAMENTAL PROBLEM THAT REMAINS UNRESOLVED
