On the Nullity of Altans and Iterated Altans

TL;DR

This paper investigates the nullity of altan and iterated altan graphs, establishing bounds and properties related to their nullity based on the parent graph and attachment sets, with implications for chemical graph theory.

Contribution

It provides sharp bounds for the nullity of altan and iterated altan graphs, including the first identification of cases with nullity excess of 2, and conjectures about convex benzenoids.

Findings

Nullity of altan equals parent nullity for odd h.

Iterated altans have the same nullity as the first.

Excess nullity of 2 occurs first in certain benzenoids with 5 hexagons.

Abstract

Altanisation (formation of the altan of a parent structure) originated in the chemical literature as a formal device for constructing generalised coronenes from smaller structures. The altan of graph $G$, denoted $\mathfrak{a}(G, H)$, depends on the choice of attachment set $H$ (a cyclic $h$-tuple of vertices of $G$). From a given pair $(G, H)$, the altan construction produces a pair $(G', H')$, where $H'$ is called the induced attachment set. Repetition of the construction, using at each stage the attachment set induced in the previous step, defines the iterated altan. Here, we prove sharp bounds for the nullity of altan and iterated altan graphs based on a general parent graph: for any attachment set with odd $h$, nullities of altan and parent are equal; for any $h$ and all $k \geq 1$, the $k$-th altan has the same nullity as the first; for any attachment set with even $h$, the nullity of the altan exceeds the nullity of the parent graph by one of the three values $\{0, 1, 2\}$. The case of excess nullity $2$ has not been noticed before; for benzenoids with the natural attachment set consisting of the CH sites, it occurs first for a parent structure with $5$ hexagons. On the basis of extensive computation, it is conjectured that in fact no altan of a convex benzenoid has excess nullity $2$.