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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]

Abstract: The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. F. Zhang, H. Li [On acyclic conjugated molecules with minimal energies, Discrete Appl. Math. 92 (1999) 71–84] characterized the trees with a perfect matching having the minimal and the second minimal energies, which solved a conjecture proposed by I. Gutman [Acyclic conjugated molecules, trees and their energies, J. Math. Chem. 1 (1987) 123–143]. In this letter, for a given positive integer we characterize the tree with the minimal energy having diameter at least . As a corollary, we also characterize the tree with the minimal Hosoya index having diameter at least . [Copyright &y& Elsevier]

The Wiener and Kirchhoff indices of a graph G are two of the most important topological indices in mathematical chemistry. A graph G is called to be a globular caterpillar if G is obtained from a complete graph Ks with vertex set {v1,v2,..., vs} by attaching ni pendent edges to each vertex vi of Ks for some positive integers s and n1,n2,...,ns, denoted by GCs;ni1s. Let GCs;n be the set of globular caterpillars GCs;ni1s with n vertices (n=s+∑i=1sni). In this article, we characterize the globular caterpillars with the minimal and maximal Wiener and Kirchhoff indices among GCs;n, respectively. [ABSTRACT FROM AUTHOR]