The wave equation, $$\ddot{u} = c^2 \Delta u,$$ can be generalized to abstract graphs by using the negative graph Laplacian in place of the physical Laplacian.

Is there a graph-theoretic analog of Perfectly Matched Layers (PML), where "outward traveling waves" exit the graph without reflection?

It is not clear to me what the definition of "outward traveling waves" should be in the graph context, but it does seem intuitively reasonable to consider nonresonant waves propagating through a small region of interest (subgraph) embedded in a much larger sparse graph.

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    $\begingroup$ Not sure about the answer here, but I guess you would need to define a set of vertices that make up the PML, thereby giving some notion of a boundary and outward traveling waves. $\endgroup$ – Doug Lipinski Dec 21 '14 at 14:37

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