Asymptotic energy of graphs

TL;DR

This paper establishes that the asymptotic energy per vertex of certain lattice graphs is invariant under specific spanning subgraph conditions, enabling unified formulas across various boundary conditions.

Contribution

It proves the invariance of asymptotic energy for lattice graphs with different boundary conditions under certain graph sequence conditions.

Findings

Asymptotic energy is the same for graphs with different boundary conditions.

Unified asymptotic energy formulas are derived for multiple lattice types.

Results apply to triangular, 3^3.4^2, and hexagonal lattices.

Abstract

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.