Addition Patterns in Carbon Allotropes:  Independence Numbers and d-Codes in the Klein and Related Graphs

The problem of predicting stoichiometries and patterns of chemical addition to a carbon framework, subject solely to the restriction that each addend excludes neighboring sites up to some distance d, is equivalent to determination of d-codes of a graph, and for d = 2 to determination of maximum independent sets. Sizes, symmetries, and numbers of d-codes are found for the all-heptagon Klein graph (prototype for “plumber's nightmare” carbon) and for three related graphs. The independence number of the Klein graph is 23, which increases to 24 for a related, but sterically relaxed, all-heptagon network with the same number of vertices and modified adjacencies. Expansion of the Klein graph and its relaxed analogue by insertion of hexagonal faces to form leapfrog graphs also allows all heptagons to achieve their maximum of 3 addends. Consideration of the π system that is the complement of the addition pattern imposes a closed-shell requirement on the adjacency spectrum, which typically reduces the size of acceptable independent sets. The closed-shell independence numbers of the Klein graph and its relaxed analogue are 18 and 20, respectively.