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The resistance distance is a novel distance function on a graph proposed by Klein and Randic [D.J. Klein and M. Randic, 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. [ABSTRACT FROM AUTHOR]

Abstract: The energy of a simple graph , denoted by , is defined as the sum of the absolute values of eigenvalues of . Let (resp., and ) be the lattice obtained from an square lattice with free boundary condition (resp., cylindrical boundary condition and the toroidal boundary condition) by adding two diagonal edges to each square. In this work, we obtain the formulae for the energies of , , and . The asymptotic energies of , and are also given. [ABSTRACT FROM AUTHOR]