  Several general results for the spectral determinant of the Schr\"odinger
operator on metric graphs are reviewed. Then, a simple derivation for the
$\zeta$-regularised 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
$\zeta$-regularised spectral determinant of the Schr\"odinger 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.
