  We study the spectral determinant of the Laplacian on finite graphs
characterized by their number of vertices V and of bonds B. We present a path
integral derivation which leads to two equivalent expressions of the spectral
determinant of the Laplacian either in terms of a V x V vertex matrix or a 2B x
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.
