Locating a double vacancy or Stone–Wales point defect on a hexagonal quantum grid

A graphene nano-ribbon structure can be modelled by a 3-regular hexagonal grid. We convert this to a rectangular coordinate system in order to identify uniquely the position of either the V2(5-8-5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {V}_2(5-8-5)$$\end{document} double vacancy (DV) defect or the Stone–Wales SW(55–77) defect. This is done by using the lengths of the closed paths along the edges of the underlying graph. By sending a signal from one of the vertices and detecting the returning impulses one can observe experimentally the spectrum of the structure. Using the trace formula it is possible to determine the lengths of all closed paths (periodic orbits) starting and ending at the given vertex where a detector is placed. We present an algorithm which enables one to pinpoint the precise coordinates of a DV defect by using at most three reference points. Similarly we provide an algorithm for finding an SW defect.