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 $Q_h \in \mathbb{R}^{N*N}$  is a matrix representation of the convolution operator by a filter function $h$ and **f** = $\mathbf{[f(t_0),f(t_2),...,f(t_{N-1}]}^\top$ is a vector representation of the function $f$

Representing convolutions with matrix multiplications

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 time shifts as multiplications by the adjacency matrix:

$(A_cf)[t]=f[(t+1)_{mod N}]$

Representing time shifts on discrete time-varying signals

The Laplacian $L_c$ for a chain graph can be represented by:

$L_c = I − A_c$

Carleton College

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

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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