Graph recognition by forming heat kernel invariants

The graph structure can be characterized using the eigenvalue spectrum of the Laplace matrix of the graph. The eigenvalue spectrum of the Laplace matrix is related to the heat equation. The derivative of the heat kernel with respect to time is determined by the Laplace matrix. The solution to the heat equation can be obtained by exponentiating the eigenvalues of the Laplace matrices with respect to time to obtain the heat kernel matrix. The heat kernel is a solution to the heat equation and represents the path length distributions on the graph. This paper investigates how invariants determined from the heat kernel can be used to determine the characteristics of graphs for the purpose of measuring graph similarity. This paper considers the characteristic of the heat kernel trace as a function of time. The heat content, i.e. the sum of the elements of the heat kernel, can be used in graph comparison for subsequent graph recognition. The heat content can be expanded into a power series with respect to time, and the coefficients of the series are calculated using the spectrum of the Laplace matrices. These coefficients are used to represent the graph structure for graph comparison. The paper presents relationships for determining the Mahalanobis distance (weighted Euclidean distance) between feature vectors - invariants of the thermal kernel and elementary symmetric polynomials of graphs that can be used for pattern recognition, computer vision tasks, signal identification tasks The aim of the work is to clarify the possibility of using the coefficients of the power series of heat content as feature vectors - characteristics of graph properties. The novelty of the methods presented in the work is that, unlike the method of spectral embedding based on the thermal kernel, this work investigates the possibility of comparing graphs (images) based on finding the Euclidean distance between invariants constructed from the coefficients of the power series of heat content. The novelty of the methods presented in the work lies in the fact that, unlike the method of spectral investment based on the thermal core, in this work, the possibility of comparing graphs (images) is investigated based on the distance of the Mahalanobis distance (weighted Euclidean distance) between the invariants built from the coefficients of the steppe of thermal maintenance.