Asymptotic incidence energy and Laplacian-energy-like invariant of the Union Jack lattice

TL;DR

This paper derives explicit formulas and asymptotic values for the incidence energy and Laplacian-energy-like invariant of the Union Jack lattice, with applications in chemical physics and graph theory.

Contribution

It provides the first closed-form and asymptotic expressions for these spectral invariants of the Union Jack lattice.

Findings

Closed-form formulas for incidence energy and Laplacian-energy-like invariant.

Asymptotic values of these invariants are explicitly calculated.

Results have potential applications in chemical physics and spectral graph theory.

Abstract

The incidence energy $\mathscr{IE}(G)$ of a graph $G$, defined as the sum of the singular values of the incidence matrix of a graph $G$, is a much studied quantity with well known applications in chemical physics. The Laplacian-energy-like invariant of $G$ is defined as the sum of square roots of the Laplacian eigenvalues. In this paper, we obtain the closed-form formulae expressing the incidence energy and the Laplacian-energy-like invariant of the Union Jack lattice. Moreover, the explicit asymptotic values of these quantities are calculated by utilizing the applications of analysis approach with the help of calculational software.