The genus of a connected orientable surface is an integer representing the maximum number of cuttings along non-intersecting simple closed curves without rendering the resultant manifold disconnected. It is equal to the number of handles on it. Alternatively, it can be defined in terms of the Euler characteristic.
The genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n handles (i.e. an oriented surface of genus n). Thus, a planar graph has genus 0, because it can be drawn on a sphere without self-crossing.
It follows directly from definitions, that the number of Laplacian eigenvalues of a graph, counting multiplicities, is equal to its number of its vertices. Therefore, the number of the Laplacian eigenvalues of the dual graph of an embedded graph is equal to the number of its faces. The number of edges in the graph and its dual is equal to half the trace of the Laplacian matrix TrL(G) or TrL(G*). It follows from the Euler identity, that one can recover the genus of the embedded graph from the eigenvalue data .
Every harmonic function on a network/graph induces direction on the edges of the graph with the current flowing from the vertex w/higher potential to the lower one.
to be continued