Abstract
We study the spectrum of the continuous Laplacian Δ on a countable connected locally finite graph Γ without self-loops, whose edges have suitable positive conductances and are identified with copies of segments [0, 1], with the condition that the sum of the weighted normal exterior derivatives is 0 at every node (Kirchhoff-type condition). In particular, we analyse the relation-between the spectrum of the operator Δ and the spectrum of the discrete Laplacian (I-P) defined on the vertices of Γ.
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Cattaneo, C. The spectrum of the continuous Laplacian on a graph. Monatshefte für Mathematik 124, 215–235 (1997). https://doi.org/10.1007/BF01298245
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DOI: https://doi.org/10.1007/BF01298245