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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]