Olga Aryasova (Inst. of Geophysics, Nat. Acad. of Sciences of Ukraine / Friedrich–Schiller–Univ. Jena)
Anna Muranova (University of Warmia)
We consider discrete normalized Laplacian for finite graphs, whose edge-weights belong to an arbitrary real-closed ordered field. We show that eigenvalues of Laplacian always belong to the same field. Moreover, estimates of the eigenvalues in terms of number of vertices as well as in terms of isoperimetric constants are presented. Further, the relation to an analogue of probability operator and the analogue of convergence rate of random walk are discussed.
As an important example we consider graphs over the Levi-Civita field, which arrise in electrical networks.