We can represent the difference operator by multiplication by the Laplacian: $$(L_cf)[t] = f[t] − f[(t + 1)_{mod N }]$$

Representing difference on discrete time-varying signals

We can represent a convolution by a filter $$h$$ as matrix multiplication on the vector **f**: $$(f \star h)(t) = \sum_{\tau=0}^{N-1} f(\tau)h(\tau - t) = Q_hf$$ where $$Q_h \in \mathbb{R}^{N \times N}$$ is a matrix representation of the convolution operator by a filter function $$h$$ and **f** = $$\mathbf{[f(t_0),f(t_1),...,f(t_{N-1})]}^\top$$ is a vector representation of the function $$f$$.

Representing Convolutions with Matrix Multiplications

The Laplacian matrix $$L_c$$ for a chain graph can be represented by the equation: $$L_c = I - A_c$$ where $$I$$ is the identity matrix and $$A_c$$ is the adjacency matrix of the chain graph.

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The adjacency matrix for a chain graph corresponds to the matrix $$A_c$$: $$ A_c[i,j] = \left\{ \begin{array}{ll} 1 & j = (i+1)_{modN} \\ 0 & \text{otherwise} \\ \end{array} \right. $$

Adjacency Matrix of a Chain Graph

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

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