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  • A curious family of convex benzenoids and their altans [Elektronski vir]
    Bašić, Nino ; Fowler, Patrick W.
    The altan graph of ▫$G$▫, ▫$\mathfrak{a}(G,H)$▫, 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 ▫$\mathfrak{a}(G,H)$▫, as kernel eigenvectors of the altan correspond to non-bonding ▫$\pi$▫ 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 ▫$\mathfrak{a}(G,H)$▫ 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 ▫$\mathbf{D}_\mathrm{3h}$▫) 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.
    Source: Discrete mathematics letters [Elektronski vir]. - ISSN 2664-2557 (Vol. 9, 2022, 111-117)
    Type of material - e-article ; adult, serious
    Publish date - 2022
    Language - english
    COBISS.SI-ID - 108528387

    Link(s):

    https://www.dmlett.com/archive/v9/DML22_v9_pp111-117.pdf
    https://doi.org/10.47443/dml.2021.s218
    DOI

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