We study the spectral determinant of the Laplacian on finite graphs characterized by their number of vertices V and bonds B. We present a path integral derivation which leads to two equivalent expressions of the spectral determinant of the Laplacian in terms of either a V_V vertex matrix or a 2B_2B link matrix that couples the arcs (oriented bonds) together. This latter expression allows us to rewrite the spectral determinant as an infinite product of contributions of periodic orbits on the graph. We also present a diagrammatic method that permits us to write the spectral determinant in terms of a finite number of periodic orbit contributions. These results are generalized to the case of graphs in a magnetic field. Several examples illustrating this formalism are presented and its application to the thermodynamic and transport properties of weakly disordered and coherent mesoscopic networks is discussed. 2000 Academic Press
1. INTRODUCTION AND MAIN RESULTS
This work is devoted to the study of the spectral properties of the Laplacian operator on finite graphs. This problem already has a long history. The properties