Formulations for diffusion processes based on graph Laplacian kernel s have been recently used to solve linear transient heat transfer problems w ith insulated boundary conditions by way of a spectral -based semi - analytical approach. This has been called the “spectral graph” (SG) approach. In this paper we show that the meshfree discretization s for corresponding peridynamic models have a similar grap h structure and lead to the same general equation as the SG approach . In this sense, the SG approach can be seen as a particular case of a peridynamic formulation. For the transient heat diffusion, we explain some differences between the SG approach and peridynamics, related to calibrat ion and discretization procedures . We use a 1D heat diffusion example to highlight some limitations the spectral -based semi -analytical method in the SG approach has compared with the direct time -integration no rmally used in computing solutions to peridynamic models. We also introduce an extension of the semi-analytical approach to diffusion problems with Dirichlet boundary conditions. Keywords: peridynamics, graph theory, graph Laplacian, spectral graph, heat transfer, diffusion 1. Introduction to Graph Laplacian exponential diffusion kernel and Peridynamics Graphs are data structure s widely used in many applications [1]. Graphs can be us ed to model complex structures, for instance, molecul es [2], proteins [3], social networks [4], image s [5], internet [6], natural language s [7]. In chemistry and biology , identify ing the function of a molecule with experimental methods can be expensive and time -consuming. Alternatively, one can use graph theory and machine learning [8] to estimate properties of the new molecule /protein based on other molecule s/proteins with similar structure s. 2 By using machine learning method s to determine which class a graph belongs to , one needs to quantify the similarity between graphs. However, due to the ir complex structure, comparing graphs is an NP-problem (nondeterministic polynomial time) [1]. To overcome the unfordable complex ity, kernel -based method s that can capture the entire structure of a graph have been used [1]. One advantage of kernel -based methods is that data is implicitly mapped into a Hilbert space and can be easily compared [9]. In this paper, we will focus on the graph Laplacian exponential diffusion kernel , which is the solution to the diffusion equation on the graph domain , obtained with a spectral -based semi -analytical approach [9][10]. The g raph Laplacian exponential diffusion kernel was first introduced in 2002 and has been used in different applications, including image classification tasks [9], community detection in networks [11], etc. In recent paper s, the graph Laplacian exponential diffusion kernel was used to solve heat diffusion in the additive manufacturing process . The additive manufactured parts are modeled as graphs and the classical heat diffusion partial diffusion equation (PDE) is solved [12][13]. In these works, the procedure was called “spectral graph (SG) approach”. Notably, until now, as remarked in [13], the SG approach has been limited to problems with insulated boundaries only, i.e., zero Neumann boundary condition s. By constructing the graph and linking each node with all other nodes within a certain distance [13], effectively a nonlocal model is created . As we shall see, at least formally, the SG approach can be se en as a particular case of a peridynamic model for diffusion , while some differences related to calibration and discretization procedures exist between them . Peridynamics was introduced as a nonlocal form of continuum mechanics in [14]. Since then, it has been successfully applied to modeling of brittle fracture [15][16], fracture and damage in composite s [17][18], heat transfer [19][20], corrosion damage [21][22], etc. In peridynamics, each material point is connected through bonds to other points, within a certain neighborhood region called ‘the horizon’. The properties of these bonds are obtained by , for example, matching the classical strain energy density under a homogeneous deformation (for elasticity) [23][24] or heat -flux under constant thermal gradient condition s [19]. Various methods can be used to discretize peridynamic formulations. The so -called “meshfree discretization”, uses a one -point Gaussian quadrature scheme for approximating the integral operator in the formulation [24]. This numerical method is popular for peridynamic models because of i ts flexibility in representing unrestricted damage evolution, which helps deliver on one of the main advantages peridynamic models have compared with classical, PDEs -based models. With the meshfree spatial discretization, the peridynamic model can be treated as a graph , in which vertices are the nodes and edges are the peridynamic bonds . The paper is organized as follows: graph concepts and graph Laplacian exponential diffusion kernel (“spectral graph approach”) are reviewed in Section 2.1. The p eridynamic diffusion model for diffusion is 3 reviewed in Section 2.2. In Section 3, using a 2D transient heat transfer problem as an example, we show that the SG approach is formally identical to a PD model solved by spectral -based semi -analytical approach. Differences related to the calibration and discretization procedures are highlighted . In Section 4, a numerical example for 1D transient heat transfer is used to show limitations of the spectral -based semi- analytical approach . In Section 5, we introduce an extension of the semi -analytical approach to diffusion problems with Dirichlet boundary conditions. In Section 6, we present conclusion s and future work. 2. The graph Laplacian expon ential diffusion kernel and the peridynamic model for diffusion 2.1. Review of g raph s and the g raph Laplacian exponential diffusion kernel In mathematics, a graph consists of a set of vertices 𝑉={𝑣1,𝑣2,…} and another set of edges 𝐸= {𝑒1,𝑒2,…} such that each edge is identified wit h an unordered pair of vertices [25]. Graphs are common ly represented as diagram s in which vertices are nodes and edges are represented by lines connecting nodes [25], as shown in Figure 1. In this paper , we stud y the weighted undirected simple graphs (no multiple edges or loops ). Although graph s can be represented as diagram s, matri ces are a convenient way to represen t graphs, especially in computational studies [25]. There are different matric es used in gra ph theory [25]; here we focus only on the adjacency and Laplacian matrices. The adjacency matrix of an 𝑁-vertex weighted undirected simple graph is an 𝑁 by 𝑁 real-symmetric matrix 𝐴=[𝑎𝑖𝑗], with 𝑎𝑖𝑗=weight of edge between node 𝑖 and node 𝑗 if there is an edge between the 𝑖𝑡ℎ and 𝑗𝑡ℎ vertices, and 𝑎𝑖𝑗=0 if not . In the weighted undirected graph in Figure 1, the vertices’ coordinates in a two-dimensional Cartesian coordinate system are shown, and the weights of edges ar e equal to the inverse of the edges length square. For example, the length of edge between vertex 𝑣1 and vertex 𝑣2 is 2.5 and the weight of this edge is 0.16. The degree matrix of an 𝑁-vertex weighted undirected simple graph is an 𝑁 by 𝑁 diagonal mat rix 𝐷= [𝑑𝑖𝑗], with 𝑑𝑖𝑖 equals the degree of 𝑖𝑡ℎ vertex which is the sum of the weight of edges connected to the 𝑖𝑡ℎ vertex. The Laplacian matrix of an 𝑁-vertices weighted undirected simple graph 𝐺 is defined as 𝐿(𝐺)= 𝐷(𝐺)−𝐴(𝐺). The matrix repres entation of the graph is shown in Figure 1. 4 Figure 1 Undirected Graph (v: vertices, e: edges) and its corresponding adjacency matrix , degree matrix, and Laplacian matrix Figure 2 A domain discretized by a uniform grid (left), and the 5 -point stencil for finite difference approximations (right). The negative Laplacian matrix can be seen as a proper discretization of the classical Laplacian operator [26][27]. For example, o n a square grid in 2D. with grid spacing Δ𝑥=Δ𝑦=ℎ, the finite difference approximation of the continuous Laplac e operator at node 𝑘, of coordinate s (𝑥𝑘,𝑦𝑘), is 𝜕2𝜃 𝜕𝑥2(𝑥𝑘,𝑦𝑘)+ 𝜕2𝜃 𝜕𝑦2(𝑥𝑘,𝑦𝑘)≈𝜃𝑙−2𝜃𝑘+𝜃𝑗 (∆𝑥)2+𝜃𝑖−2𝜃𝑘+𝜃𝑚 (Δ𝑦)2=1 ℎ2(𝜃𝑙+𝜃𝑗+𝜃𝑖+𝜃𝑚−4𝜃𝑘). If the square grid is seen as a graph, node 𝑘 is connected to four neighboring nodes (node 𝑖, node 𝑗, node 𝑙, and node 𝑚). In the graph, the edge 𝑣1(0.5,4) 𝑣2(3,4) 𝑣3(5,3) 𝑣4(4,1) 𝑣5(1,1) 𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 𝑂 𝑦 𝑥 𝐴= [ 00.1600.04710.1081 0.1600.20 0 00.200.20.05 0.0471 00.20 0 0.1081 00.050 0] 𝐷= [ 0.3152 000 0 00.3600 0 000.450 0 0000.2471 0 00000.1581] 𝐿= [ 0.3152−0.160−0.0471−0.1081 −0.160.36−0.20 0 0−0.20.45−0.2−0.05 −0.0471 0−0.20.2471 0 −0.1081 0−0.0500.1581] node 𝑘 (𝑥𝑘,𝑦𝑘) node 𝑙 (𝑥𝑙,𝑦𝑙) node 𝑗 (𝑥𝑗,𝑦𝑗) node 𝑖 (𝑥𝑖,𝑦𝑖) node 𝑚 (𝑥𝑚,𝑦𝑚) 5 weight is the inverse of edge length square , same as Figure 1. The continuous Laplacian at node 𝑘 can then be approximated by the negative Laplacian matrix multi plied by vector 𝜽, which leads to: 1 ℎ2(𝜃𝑙+𝜃𝑗+𝜃𝑖+𝜃𝑚−4𝜃𝑘), (see Figure 3). Thus, −𝐿 is the same as the finite difference approximation of the continuous Laplacian operator [9]. In the limit ℎ→0, this approximati on becomes exact [9]. Figure 3 The approximat ion of the continuous Laplacian at node k using the Laplacian matrix In some recent publi cations [12][13], the spectral graph theory (which studies eigenvectors of matrices of graphs, e.g. adjacent matrix and Laplacian matrix ) is used to predict heat diffusion in parts built by additi ve manufacturing processes . In the se