On a construction using commuting regular graphs

We consider a construction using a pair of commuting regular graphs that generalizes the constructions of the middle, total, and quasitotal graphs. We derive formulae for the characteristic polynomials of the adjacency and Laplacian matrices and for the Ihara zeta function of the resulting graph. Using these formulae, we express the number of spanning trees and the Kirchhoff index of the resulting graph in terms of the Laplacian spectra of the two regular graphs used in the construction.