A Curious Family of Convex Benzenoids and Their Altans

Abstract

The altan graph of G , a ( G, H ) , is constructed from graph G by choosing an attachment set H from the vertices of G and attaching vertices of H to alternate vertices of a new perimeter cycle of length 2 | H | . When G is a polycyclic plane graph with maximum degree 3 , the natural choice for the attachment set is to take all perimeter degree- 2 vertices in the order encountered in a walk around the perimeter. The construction has implications for the electronic structure and chemistry of carbon nanostructures with molecular graph a ( G, H ) , as kernel eigenvectors of the altan correspond to non-bonding π molecular orbitals of the corresponding unsaturated hydrocarbon. Benzenoids form an important subclass of carbon nanostructures. A convex benzenoid has a boundary on which all vertices of degree 3 have exactly two neighbours of degree 2 . The nullity of a graph is the dimension of the kernel of its adjacency matrix. The possible values for the excess nullity of a ( G, H ) over that of G are 2 , 1 , or 0 . Moreover, altans of benzenoids have nullity at least 1 . Examples of benzenoids where the excess nullity is 2 were found recently. It has been conjectured that the excess nullity when G is a convex benzenoid is at most 1 . Here, we exhibit an infinite family of convex benzenoids with 3 -fold dihedral symmetry (point group D 3h ) where nullity increases from 2 to 3 under altanisation. This family accounts for all known examples with the excess nullity of 1 where the parent graph is a singular convex benzenoid.