关于Altans与迭代Altans的零性
On the Nullity of Altans and Iterated Altans
摘要：
Altanisation（形成母体结构的Altan）起源于化学文献，作为从较小结构构建广义晕苯的正式方法。图$G$的代数，记为$\mathfrak{a}(G, H)$，取决于附着集$H$（$G$的循环$h$ -顶点元组）的选择。从给定的$(G, H)$对，Altan结构产生一对$(G', H')$，其中$H'$被称为诱导连接集。重复构造，在每个阶段使用上一步中导出的附件集，定义迭代的altan。本文证明了基于一般父图的Altan图和迭代Altan图的零度的精确界：对任意具有奇数$h$的附着集，Altan图和父图的零度相等;对任意$h$和所有$k \geq 1$，第$k$ Altan图和第一个图的零度相同;对于具有偶数$h$的任何连接集，Altan的空度超过父图的空度三个值之一$\{0, 1, 2\}$。$2$的过量无效性的情况以前没有被注意到;对于具有由CH位点组成的天然连接集的benzoline，它首先发生在具有$5$六边形的母体结构中。在大量计算的基础上，证明了凸苯型分子的所有代数都没有多余的零元$2$。
英文摘要：
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 $2$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
