Abstract: 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 heat content of the thermal kernel is used when comparing graphs for their recognition. 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 heat 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 embed-ding based on the heat 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 heat kernels, 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 expanding of heat content. A method for learning the Mahalanobis matrix from the Laplace matrices of graphs is proposed. The novelty of the methods presented in the work is that the possibility of comparing graphs (images) based on the Mahalanobis distance (weighted Euclidean distance) between invariants constructed using the coefficients of the heat content matrix is investigated. A method for training the Mahalanobis matrix using the Laplace matrices of graphs is proposed.

Cite: Chukanov S. , Chernukha I.
Comparison of graphs based on the formation of heat kernel invariants
In compilation "Dynamics of Systems, Mechanisms and Machines" (Dynamics-2025). – IEEE., 2025. DOI: 10.1109/dynamics68764.2025.11302150 OpenAlex

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W7117104710

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