The Kirchhoff indices of join networks

In this work we compute the effective resistance between any pair of vertices with respect to a value  λ≥0λ≥0and a weight  ωω on the vertex set of the join network   in terms of the corresponding parameters for the factors. In particular, this allows us to express the Kirchhoff index with respect to a value λ≥0λ≥0 and a weight ωω of the join network in terms of the Kirchhoff indices of the factors. Moreover, we obtain a full description of the eigenvalues and the corresponding eigenfunctions for a positive semi-definite Schrödinger operator on a join network. Finally, we compute the effective resistances and the generalized Kirchhoff index with respect to a value λ≥0λ≥0 and a weight ωω for some families of join networks, specifically for star and cone networks.