The paper introduces a family of kernels on graphs based on the concept of regularization operators, generalizing the notion of regularization and Green's functions used for real-valued functions to graphs. The authors show that diffusion kernels can be derived from this framework and that positive, monotonically decreasing functions on the unit interval lead to kernels and corresponding regularization operators. The paper places earlier results on graph kernels, such as those by Kondor and Lafferty and Belkin and Niyogi, within the framework of Regularization Theory. It defines the graph Laplacian and its properties, relating it to the Laplace operator on real-valued functions, and discusses the spectrum of the normalized Laplacian for regular graphs. The paper also explores the analogy between the Laplacian and the Laplace operator, highlighting how both quantify local variations or smoothness of functions.The paper introduces a family of kernels on graphs based on the concept of regularization operators, generalizing the notion of regularization and Green's functions used for real-valued functions to graphs. The authors show that diffusion kernels can be derived from this framework and that positive, monotonically decreasing functions on the unit interval lead to kernels and corresponding regularization operators. The paper places earlier results on graph kernels, such as those by Kondor and Lafferty and Belkin and Niyogi, within the framework of Regularization Theory. It defines the graph Laplacian and its properties, relating it to the Laplace operator on real-valued functions, and discusses the spectrum of the normalized Laplacian for regular graphs. The paper also explores the analogy between the Laplacian and the Laplace operator, highlighting how both quantify local variations or smoothness of functions.