Spectral determinant on quantum graphs

We study the spectral determinant of the Laplacian on finite graphs charact erized by their number of vertices V and bonds B. We present a path integra l derivation which leads to two equivalent expressions of the spectral dete rminant of the Laplacian in terms of either a V x V vertex matrix or a 2B x 2B link matrix that couples the areas (oriented bonds) together, This latt er 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 t erms of a finite number of periodic orbit contributions. These results are generalized to the case of graphs in a magnetic field, Several examples ill ustrating this formalism are presented and its application to the thermodyn amic and transport properties of weakly disordered and coherent mesoscopic networks is discussed. (C) 2000 Academic Press.