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.

Laplacian Matrix of a Chain Graph

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

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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 $$f(t)$$.

Chain Graph Representation of Discrete Time-Varying Signals

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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