The Kirchhoff Index of Toroidal Meshes and Variant Networks

The resistance distance is a novel distance function on electrical network theory proposed by Klein and Randić. The Kirchhoff index Kf(G) is the sum of resistance distances between all pairs of vertices in G. In this paper, we established the relationships between the toroidal meshes network Tm×n and its variant networks in terms of the Kirchhoff index via spectral graph theory. Moreover, the explicit formulae for the Kirchhoff indexes of L(Tm×n), S(Tm×n), T(Tm×n), and C(Tm×n) were proposed, respectively. Finally, the asymptotic behavior of Kirchhoff indexes in those networks is obtained by utilizing the applications of analysis approach