Physica A 422 (2015) 193–202 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa Asymptotic incidence energy of lattices Jia-Bao Liu a,b , Xiang-Feng Pan a,∗ a School of Mathematical Sciences, Anhui University, Hefei 230601, PR China b Department of Public Courses, Anhui Xinhua University, Hefei 230088, PR China highlights • We propose the incidence energy per vertex problem for lattice systems. • The explicit asymptotic values of IE (G) for various lattices are obtained. • We deduce IE (G) of many types of lattices is independent of various boundary conditions. article info Article history: Received 12 May 2014 Received in revised form 23 October 2014 Available online 12 December 2014 Keywords: Lattice Energy Incidence energy Laplacian spectrum abstract The energy of a graph G arising in chemical physics, denoted by E (G), is defined as the sum of the absolute values of the eigenvalues of G. As an analogue to E (G), the incidence energy IE (G), defined as the sum of the singular values of the incidence matrix of G, is a much studied quantity with well known applications in chemical physics. In this paper, based on the results by Yan and Zhang (2009), we propose the incidence energy per vertex problem for lattice systems, and present the closed-form formulae expressing the incidence energy of the hexagonal lattice, triangular lattice, and 3 3 .4 2 lattice, respectively. Moreover, we show that the incidence energy per vertex of lattices is independent of the toroidal, cylindrical, and free boundary conditions. In particular, the explicit asymptotic values of the incidence energy in these lattices are obtained by utilizing the applications of analysis approach with the help of calculational software. © 2014 Elsevier B.V. All rights reserved. 1. Introduction Lattices have several attractive features that make them interesting candidates for use in matter physics. The resistance distances of lattices were well studied on the basis of electrical network theory in Refs. [1–3]. A general problem of interest in physics, chemistry and mathematics is the calculations of various energies of lattices [2,4–6], since the energies can be used to estimate the total π -electron energy in conjugated hydrocarbons [7]. Another energy application is dimer problem in statistical physics, the problem considers the molecular free energy per dimer, which mimics the adsorption of diatomic molecules on a surface [4]. Historically in lattice statistics, the hexagonal lattice, triangular lattice, and 3 3 .4 2 lattice have attracted the most attention [1,4,8–11]. Throughout the paper all graphs considered are simple and undirected. Let A(G) be the adjacency matrix, the eigenvalues of G are the eigenvalues of its adjacency matrix A(G) [12]. These eigenvalues, arranged in a non-increasing order, will be denoted as λ 1 (G), λ 2 (G),...,λ n (G). In 1976 Gutman introduced the concept of energy E (G) [7] for a simple graph G, which is defined as E (G) =  n i=1 |λ i (G)|. ∗ Corresponding author. Tel.: +86 551 63861313. E-mail address: xfpan@ahu.edu.cn (X.-F. Pan). http://dx.doi.org/10.1016/j.physa.2014.12.006 0378-4371/© 2014 Elsevier B.V. All rights reserved.