Recovering Laplacian Lattices from $L$-Functions of Graphs

Abstract

We introduce $L$-functions associated with characters of the Jacobian of a finite graph, as a graph-theoretic analogue of the $L$-functions arising from unramified coverings of algebraic curves. These $L$-functions are defined using the Riemann--Roch structure on the graph and extend Lorenzini's two-variable zeta function. We show that if two connected graphs without bridges have isomorphic Jacobians and their $L$-functions agree under the induced correspondence of characters, then their Laplacian lattices coincide. We also show that these $L$-functions satisfy a functional equation. Finally, we give examples showing that neither the Jacobian nor the Lorenzini zeta function alone determines the Laplacian lattice.

arXiv commentTo be published in Bulletin of the Brazilian Mathematical Society