This paper proposes a general method for constructing kernels over discrete structures, particularly graphs, using matrix exponentiation. The authors introduce a class of exponential kernels called diffusion kernels, which are based on the heat equation and can be seen as a discretization of the Gaussian kernel in Euclidean space. These kernels are designed to capture long-range relationships in data that are naturally represented as graphs, such as document networks, social networks, and chemical structures. The paper discusses the mathematical properties of diffusion kernels, including their positive semi-definiteness and symmetry, and provides interpretations in terms of stochastic processes, electrical circuits, and spectral graph theory. It also explores how diffusion kernels can be computed for specific graph structures, such as $k$-regular trees, complete graphs, and closed chains. Preliminary experiments on UCI datasets show that diffusion kernels can effectively classify categorical data, even when standard Euclidean kernels are not applicable. The authors conclude that diffusion kernels offer a natural approach to capturing intrinsic structures in data without the need to embed them into Euclidean space, making them useful for a wide range of applications.This paper proposes a general method for constructing kernels over discrete structures, particularly graphs, using matrix exponentiation. The authors introduce a class of exponential kernels called diffusion kernels, which are based on the heat equation and can be seen as a discretization of the Gaussian kernel in Euclidean space. These kernels are designed to capture long-range relationships in data that are naturally represented as graphs, such as document networks, social networks, and chemical structures. The paper discusses the mathematical properties of diffusion kernels, including their positive semi-definiteness and symmetry, and provides interpretations in terms of stochastic processes, electrical circuits, and spectral graph theory. It also explores how diffusion kernels can be computed for specific graph structures, such as $k$-regular trees, complete graphs, and closed chains. Preliminary experiments on UCI datasets show that diffusion kernels can effectively classify categorical data, even when standard Euclidean kernels are not applicable. The authors conclude that diffusion kernels offer a natural approach to capturing intrinsic structures in data without the need to embed them into Euclidean space, making them useful for a wide range of applications.