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تسجيل مستخدم جديدAsymptotic energy of graphs
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The energy of a simple graph $G$ arising in chemical physics, denoted by $mathcal E(G)$, 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 ${G_n}$ is a sequence of finite simple graphs with bounded average degree and ${G_n}$ a sequence of spanning subgraphs of ${G_n}$ such that almost all vertices of $G_n$ and $G_n$ have the same degrees, then $G_n$ and $G_n$ 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, $3^3.4^2$, and hexagonal lattices with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions simultaneously.
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