化学螺母图空间的作图
Charting the space of chemical nut graphs
摘要：
不饱和碳骨架的分子图或除去氢原子的碳氢化合物的分子图是化学图。化学图是最大度为$3$或更小的连通简单图，坚果图是具有奇异邻接矩阵的连通简单图，奇异邻接矩阵具有一个零特征值和一个不含零元素的非平凡核特征向量。坚果图没有度为$1$的顶点：它们是无叶的。这两个集合的交集，化学坚果图，在化学和分子物理中的应用是有兴趣的，对应于完全分布的自由基反应性和omniconducting行为在费米能级的结构。一个化学螺母图由度为$2$的$v_2 \ge 0$顶点和度为$3$的偶数个顶点$v_3 > 0$组成。借助系统的局部结构，产生较大的螺母图从较小的，对应于可实现的化学螺母图的组合$(v_3, v_2)$的特点。除了有限的一组小的情况下，和两个简单定义的无穷级数，所有的组合$(v_3, v_2 )$与$v_3 > 0$的偶数值是可实现的化学螺母图。在这些组合中，只有$(20,0)$不能通过平面化学螺母图实现。主要结果刻画了所有阶$n$的化学螺母图的边数的范围。
英文摘要：
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 $n$.
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。
