Kernel and trace formula for the exponential of the Laplace-Beltrami operator on a decorated graph

For the kernel of the Laplace operator Delta(Lambda) with potential Sigma (k)(j=1) c(j) delta (qj) (x) on a manifold, (the operator is given by a Lagrangian plane Lambda subset of C-k circle plus C-k supercript stop), an isomorphism Gamma: ker Delta(Lambda) --> Lambda boolean AND L is described, where L is a special Lagrangian plane (whose explicit form is evaluated). A similar assertion holds for the Laplace operator Delta(Lambda 0) on a decorated graph; for such a graph (obtained by decorating a connected finite graph with n edges and v vertices) with "continuity" conditions, the inequality 1 <= dim ker Delta(Lambda 0) <= n - v + 2 is obtained. It is also proved that the quantity n - v + 1-dim ker Delta(Lambda 0) cannot reduce when adding new edges and manifolds. The first terms of the expansion of Tr(exp(-tH(Lambda))) are found.