A curious family of convex benzenoids and their altans

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 D3h) 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.