Resistance distances and the Kirchhoff index in Cayley graphs

In this paper, closed-form formulae for the Kirchhoff index and resistance distances of the Cayley graphs over finite abelian groups are derived in terms of Laplacian eigenvalues and eigenvectors, respectively. In particular, formulae for the Kirchhoff index of the hexagonal torus network, the multidimensional torus and the tt-dimensional cube are given, respectively. Formulae for the Kirchhoff index and resistance distances of the complete multipartite graph are obtained.

► Closed-form formulae for the Kirchhoff index and resistance distances of the Cayley graphs over finite abelian groups are obtained.
► Formulae for the Kirchhoff index of the hexagonal torus network, the multidimensional torus and the tt-dimensional cube are derived.
► Formulae for the Kirchhoff index and resistance distances of the complete multipartite are given.