The asymptotic behavior of (degree-)Kirchhoff indices of iterated total graphs of regular graphs

Let G be a simple connected graph. Denote the Kirchhoff index and the degree-Kirchhoff index of G by Kf(G) and Kf∗(G), respectively. This paper considers the asymptotic behavior of Kf(Tk(G)) and Kf∗(Tk(G)) of the iterated total graph Tk(G) of an r-regular graph G. We show that the asymptotic behavior of these indices is independent of the structure of G and only dependent on the degree and the number of vertices of G.