The adjacency matrix for a chain graph corresponds to the matrix $A_c$:

$$ A_c[i,j] =   \left\{
\begin{array}{ll}
      1 & j = (i+1)_{modN} \\
      0 & otherwise \\
\end{array} 
\right.  $$

Adjacency Matrix of Chain Graph

Given a discrete time-varying signal $$f(t_0), f(t_1), \dots, f(t_{N-1})$$, we can represent this signal as a chain (or cycle) graph. Each node in the graph represents a point in time $$t$$ and the signal value at that node represents the function value $$f(t)$$.

Carleton College

Google

 In this perspective, where GNNs were developed as a generalization of Euclidean convolutions to the non-Euclidean graph domain, they first arise from the question: How can we generalize the notion of convolutions to general graph-structured data?

GNN and Graph Convolutions

Hamilton, W.L. Graph Representation Learning. *Synthesis Lectures on Artificial Intelligence and Machine Learning*. 2020, 14(3), 1–159. https://doi.org/10.2200/S01045ED1V01Y202009AIM046

Graph Representation Learning by William Hamilton

In order to generalize the notion of a convolution to graphs, we first must define
what we wish to generalize and provide some brief background details. We can define the general continuous convolution operation * shown below, let f and h be two functions.


 The General Continuous Convolution Operation

There’re several types of GNN inspired by convolutions operations and their conenctions:

1)	GNN That Use Purely Convolutional Approaches

2)	Graph Convolutional Networks (GCN)

3)	GNN Without Message Passing


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