Summary: The authors compute explicitly the heat kernel on the surface of cones as well as on their generalizations. A procedure similar to the Fourier transform is employed in order to combine two Green’s functions: one for the Bessel equation on the positive half-line and another for the Laplacian on graph networks. An analogue of the Poisson summation formula is derived from the residue theorem applied to the Green’s function. Numerical computations are also implemented to determine some geometric quantity via the asymptotic expansion of the spectral function as \(t\) goes to zero.