[orange3xchicken]

Just re spectral methods on graphs: There are a number of really interesting interpretations of spectral methods including energy stabilization of spring networks and discrete analogues to heat diffusion over grids. The area is really fascinating.

One general interpretation is that the eigenvalues of the adjacency & other matrices that capture the connectivity of the graph are solutions to optimization problems [0]. Usually, these optimization problems and their solutions correspond to relaxations of interesting combinatorial features or operations on graphs - e.g. different ways of cutting graphs to yield binary partitions on the nodes (spectral clustering [1]).

[0] https://en.wikipedia.org/wiki/Min-max_theorem

[1] https://en.wikipedia.org/wiki/Spectral_clustering