Asymptotic energy of graphs
Abstract
The energy of a simple graph G arising in chemical physics, denoted by E(G), is defined as the sum of the absolute values of eigenvalues of G. We consider the asymptotic energy per vertex (say asymptotic energy) for lattice systems. In general for a type of lattice in statistical physics, to compute the asymptotic energy with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions are different tasks with different hardness. In this paper, we show that if \Gn\ is a sequence of finite simple graphs with bounded average degree and \Gn'\ a sequence of spanning subgraphs of \Gn\ such that almost all vertices of Gn and Gn' have the same degrees, then Gn and Gn' have the same asymptotic energy. Thus, for each type of lattices with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions, we have the same asymptotic energy. As applications, we obtain the asymptotic formulae of energies per vertex of the triangular, 33.42, and hexagonal lattices with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions simultaneously.