A curious family of convex benzenoids and their altans | [Discrete Mathematics Letters • 2022]

Author(s):

1. Nino Basic: FAMNIT (Faculty of Mathematics, Natural Sciences and Information Technologies), University of Primorska, Koper, Slovenia;IAM (Inˇstitut Andrej Maruˇsiˇc), University of Primorska, Koper, Slovenia ;Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

2. Patrick W. Fowler: Department of Chemistry, University of Sheffield,Sheffield,UK

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 2jHj. 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 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.

Page(s): 111-117

DOI: 10.47443/dml.2021.s218

Published: Journal: Discrete Mathematics Letters, Volume: 9, Issue: 0, Year: 2022

Keywords:

convex benzenoid , benzenoid , altan , chemical graph , pointgroup symmetry , nullity

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