In graph theory, a field within mathematics, there is a problem of determining the genus of a graph from its Dirichlet-to-Neumann map and the eigenvalues of the Laplace–Beltrami operator. The main tools used are a maximum principle, some determinant identities and a variation-diminishing property.
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.
to be continued
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