Topological energy of the distance matrix

Graph energy and entropy are closely related to the network structure, and there are various types of definitions. An extended issue is how to define energy on a point set with a metric structure. This article provides the graph energy defined on the distance matrix. We use the order complex in topological data analysis to filter the distance matrix and generate a graph sequence. Then, the energy sequence is defined by the graph sequence. We calculate the average in the appropriate interval to get the definition of the topological energy of distance matrix. We also analyzed the relationship between topological energy and entropy. Finally, this article provides examples of financial market and chaotic systems. Calculations show that the extended energy can capture the changes in the point set caused by the nonlinear coordinate transformation. The method proposed here provides an indicator of a distance matrix, which makes it possible to observe a point set with a metric structure from the perspective of graph energy. (C) 2021 Elsevier B.V. All rights reserved.