### Abstract

The classic paper of Roth shows that, for a graph with the Kirchhoff boundary conditions, the constant term in the heat-kernel expansion equals half the difference between the vertex number and the edge number. An index interpretation is achieved by comparing with another operator on the same graph with different boundary conditions. The construction is a generalization of Peter Gilkey's treatment of Neumann and Dirichlet boundary conditions on the interval as the simplest case of the de Rham complex on a manifold with boundary. It uses Justin Wilson's generalization of Roth's formula to any boundary conditions that yield a real bond-to-bond S-matrix.