论文标题
绘制化学螺母图的空间
Charting the space of chemical nut graphs
论文作者
论文摘要
化学图是不饱和碳框架或碳氢化合物的分子图。化学图是一个连接的简单图，最高度$ 3 $或更少。螺母图是一个连接的简单图，具有单数邻接矩阵，其特征值为零和一个没有零条目的非平凡核特征向量。坚果图没有$ 1 $的顶点：它们是无叶的。这两组的交点是化学螺母图，在化学和分子物理学中具有应用，对应于在费米水平上具有完全分布的自由基反应性和量化行为的结构。一个化学螺母图由$ v_2 \ ge 0 $ $ 2 $的顶点和一个均匀的数字，$ v_3> 0 $，$ 3 $。借助系统的局部结构，这些结构从较小的较小的螺母图中产生较大的螺母图，表征了与可实现的化学螺母图相对应的组合$（v_3，v_2）$。除了一组有限的小型案例和两个简单定义的无限序列外，所有组合$（v_3，v_2）$，均匀为$ v_3> 0 $，可以将其视为化学螺母图。在这些组合中，只有$（20,0）$无法通过平面化学螺母图实现。主要结果是所有订单的化学螺母图$ n $的边缘计数范围。
Molecular graphs of unsaturated carbon frameworks or hydrocarbons pruned of hydrogen atoms, are chemical graphs. A chemical graph is a connected simple graph of maximum degree $3$ or less. A nut graph is a connected simple graph with a singular adjacency matrix that has one zero eigenvalue and a non-trivial kernel eigenvector without zero entries. Nut graphs have no vertices of degree $1$: they are leafless. The intersection of these two sets, the chemical nut graphs, is of interest in applications in chemistry and molecular physics, corresponding to structures with fully distributed radical reactivity and omniconducting behaviour at the Fermi level. A chemical nut graph consists of $v_2 \ge 0$ vertices of degree $2$ and an even number, $v_3 > 0$, of vertices of degree $3$. With the aid of systematic local constructions that produce larger nut graphs from smaller, the combinations $(v_3, v_2)$ corresponding to realisable chemical nut graphs are characterised. Apart from a finite set of small cases, and two simply defined infinite series, all combinations $(v_3, v_2 )$ with even values of $v_3 > 0$ are realisable as chemical nut graphs. Of these combinations, only $(20,0)$ cannot be realised by a planar chemical nut graph. The main result characterises the ranges of edge counts for chemical nut graphs of all orders $n$.
