Asymptotic incidence energy of lattices

• We propose the incidence energy per vertex problem for lattice systems.
• The explicit asymptotic values of IE(G)IE(G) for various lattices are obtained.
• We deduce IE(G)IE(G) of many types of lattices is independent of various boundary conditions.

The energy of a graph GG arising in chemical physics, denoted by E(G)E(G), is defined as the sum of the absolute values of the eigenvalues of GG. As an analogue to E(G)E(G), the incidence energy IE(G)IE(G), defined as the sum of the singular values of the incidence matrix of GG, 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 33.4233.42 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.