In this work, we explicitly compute the group inverse of symmetric and periodic Jacobi matrices with constant elements that have been extended by adding a row and a column conveniently de ned. For this purpose, we interpret such matrices as the combinatorial Laplacian of a non-complete wheel that has been obtained by adding a vertex to a cycle and some edges conveniently chosen. The obtained group inverse is an incomplete block matrix with a block Toeplitz structure. In addition, we obtain the e ffective resistances and the Kirchhoff index of non-complete wheels.