This paper describes a graph-spectral method for simplifying the structure of a graph. Our starting point is the lazy random walk on the graph, which is determined by the heat-kernel of the graph and can be computed from the spectrum of the graph Laplacian. We characterise the random walk using the commute time between nodes, and show how this quantity may be computed from the Laplacian spectrum using the discrete Green's function. Our idea is to augment the graph with an auxiliary node which acts as a heat source. We use the pattern of commute times from this node to decompose the graph into a sequence of layers. These layers can be located using the Green's function. We exploit this decomposition to develop a layer-by-layer graph-matching strategy. The matching method uses the commute time from the auxiliary node as a node-attribute.
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