Determining the defect type in a graphene lattice

We consider four common defects that appear in an infinite single-layer graphene lattice, i.e. single vacancy V1(5−9), Stone-Wales SW (55−77), and double vacancies V2(5−8−5) and V2(555−777). Investigating the honeycomb pattern of the graphene, we make use of the concept of shortest closed paths (periodic orbits) in the underlying topological structure. Using properties of the shortest closed paths of odd length we present and prove mathematically an algorithm that classifies which one of these defects occurs.