Recently, Smilansky expressed the determinant of the bond scattering matrix of a graph by means of the determinant of its Laplacian. We present another proof for this Smilansky’s formula by using some weighted zeta function of a graph. Furthermore, we reprove a weighted version of Smilansky’s formula by Bass’ method used in the determinant expression for the Ihara zeta function of a graph.

Given a vertex-weighted oriented graph, we can associate to it set of monomials. We consider the toric ideal whose defining map is given by these find generating for certain classes graphs which depends on combinatorial structure and weights graph. provide result analogous unweighted, unoriented graph case, show that when associated simple has only trivial even closed walks, zero ideal. Moreove...