ζ-regularized spectral determinants on metric graphs

Several general results for the spectral determinant of the Schroedinger operator on metric graphs are reviewed. Then, a simple derivation for the ζ-regularized spectral determinant is proposed, based on the Roth trace formula. Two types of boundary conditions are studied: functions continuous at the vertices and functions whose derivative is continuous at the vertices. The ζ-regularized spectral determinant of the Schroedinger operator acting on functions with the most general boundary conditions is conjectured in conclusion. The relation to the Ihara, Bass and Bartholdi formulae obtained for combinatorial graphs is also discussed.