Classification of k-defect holes on a graphene sheet

Holey graphene sheet is a promising nano-scaled material with a number of applications for electrochemical energy storage and gas sensors. Further, the joining of a carbon nanotube on a holey flat graphene creates new 3D material combining remarkable mechanical properties from both materials. In order to join a carbon nanotube with defect atoms on its end, we need a hole in graphene sheet with an equal number of defect atoms on its border. A k-defect hole on a graphene sheet is defined to be a simply-connected region that has k carbon atoms on the edge of the hole with one missing bond. In this study, a mathematical technique is presented to enumerate and classify all possible k-defect holes on the graphene sheet. The predicted joining structure for each hole formation is also determined. We introduce the least square and the discrepancy variance parameters that assess the hole's joining quality in order to choose the best k-defect holes to join with a carbon nanotube.