On the Kirchhoff index of some toroidal lattices

Abstract

The resistance distance is a novel distance function on a graph proposed by Klein and Randić [D.J. Klein and M. Randić, Resistance distance, J. Math. Chem. 12 (1993), pp. 81–85]. The Kirchhoff index of a graph G is defined as the sum of resistance distances between all pairs of vertices of G. In this article, based on the result by Gutman and Mohar [I. Gutman and B. Mohar, The quasi-Wiener and the Kirchhoff indices coincide, J. Chem. Inf. Comput. Sci. 36 (1996), pp. 982–985], we compute the Kirchhoff index of the square, 8.8.4, hexagonal and triangular lattices, respectively.

Keywords:

• Kirchhoff index
• lattice
• Laplacian eigenvalue

AMS Subject Classification::

• Primary 05C15
• 05C16

Acknowledgements

The author is grateful to the referee for providing some helpful revising suggestions, who also provided the references 3,5,6,8. The author was supported in part by NSFC Grant(10771086).