Graph Laplacians and topology

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Schrödinger operators on graphs and geometry I: Essentially bounded potentials (2008, methodology): “To calculate the Euler characteristic the following formula has been derived (see (4.1) from [ 26 ]):”
Rayleigh estimates for differential operators on graphs (2014, background): “Differential equations on metric graphs are also interesting from purely mathematical point of view and exhibit unusual spectral phenomena; see [6], [15], [24], and [25].”