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

$L_c = I − A_c$

Laplacian for a Chain Graph

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

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

Carleton College

Given a discrete time-varying signal $f(t_0),f(t_2),...,f(t_{N−1})$, 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 $f(t)$

Representing discrete time-varying signals with a 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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