Divergences in quantum electrodynamics on a graph

We consider a model of quantum electrodynamics (QED) on a graph as the generalization of dimensional deconstruction with the Abelian symmetry. Arbitrary structures of the theory space correspond to the graphs consisting of vertices and edges. The mass spectrum of the model is expressed in terms of eigenvalues of the Laplacian for the graph. We also find that physical massless scalar modes are associated with the fundamental tie set matrix on the graph. We further investigate the one-loop divergences in the model by use of the background field method. (c) 2005 American Institute of Physics.