The Eigenvalues of the Laplacian on Locally Finite Networks Under Generalized Node Transition

We consider the canonical continuous Laplacian on an infinite locally finite network with different edge lengths under natural transition conditions as continuity at the ramification nodes and Kirchhoff flow conditions at all vertices that can be consistent with the Laplacian or not. It is shown that the eigenvalues of this Laplacian in a L∞-setting are closely related to those of a row–stochastic operator of the network resulting from the length adjacency operator and the weights in the Kirchhoff condition. In this way the point spectrum in L∞ is determined completely in terms of combinatorial and metrical quantities of the underlying graph as in the finite case [2] and as in the equal length case [10], and, in addition, in terms of the coefficients in the transition condition. Though the multiplicity formulae are formally the same as in the cited cases, the multiplicities can change strongly. Another main concern is the real or nonreal character of the eigenvalues.