Bounds for the Kirchhoff index of regular graphs via the spectra of their random walks

Using probabilistic tools, we give tight upper and lower bounds for the Kirchhoff index of any d-regular N-vertex graph in terms of d, N, and the spectral gap of the transition probability matrix associated to the random walk on the graph. We then use bounds of the spectral gap of more specialized graphs, available in the literature, in order to obtain upper bounds for the Kirchhoff index of these specialized graphs. As a byproduct, we obtain a closed-form formula for the Kirchhoff index of the d-dimensional cube in terms of the first inverse moment of a positive binomial variable.