[论文解读] On the arithmetic of graphs

Oliver Knill|arXiv (Cornell University)|Jun 19, 2017

Topological and Geometric Data Analysis参考文献 11被引用 7

一句话总结

本文通过三种环结构——弱积、直积和强积——引入并分析了有限单图的算术运算，证明强环同伦于斯坦利-雷伊纳尔德笛卡尔环，从而可应用库恩尼茨公式，并确认欧拉示性数与团数是环同态。此外，还表明连接拉普拉斯算子在强积下张量化，保持谱与上同调兼容性。

ABSTRACT

The Zykov ring of signed finite simple graphs with topological join as addition and compatible multiplication is an integral domain but not a unique factorization domain. We know that because by taking graph complements, it becomes isomorphic to the strong Sabidussi ring with disjoint union as addition. We prove that the Euler characteristic is a ring homomorphism from the strong ring to the integers by demonstrating that the strong ring is homotopic to a Stanley-Reisner Cartesian ring. More generally, the Kuenneth formula holds on the strong ring so that the Poincare polynomial is compatible with the ring structure. The Zykov ring has the clique number as a ring homomorphism. Furthermore, the Cartesian ring has the property that the functor which attaches to a graph the spectrum of its connection Laplacian is multiplicative. The reason is that the connection Laplacians do tensor under multiplication, similarly to what the adjacency matrix does for the weak ring. The strong ring product of two graphs contains both the weak and direct product graphs as subgraphs. The Zykov, Sabidussi or Stanley-Reisner rings are so manifestations of a network arithmetic which has remarkable cohomological properties, dimension and spectral compatibility but where arithmetic questions like the complexity of detecting primes or factoring are not yet studied well. We illustrate the Zykov arithmetic with examples, especially from the subring generated by point graphs which contains spheres, stars or complete bipartite graphs. While things are formulated in the language of graph theory, all constructions generalize to the larger category of finite abstract simplicial complexes.

研究动机与目标

• 通过弱积、直积和强积三种不同的环结构，建立图算术的严格代数框架。
• 证明图的强环同伦于斯坦利-雷伊纳尔德笛卡尔环，从而实现上同调与谱兼容性。
• 证明欧拉示性数与团数是从强环和齐科夫环到整数的环同态。
• 证明两个图的强积的连接拉普拉斯算子等于其各自连接拉普拉斯算子的张量积。
• 研究图积的谱与上同调性质，特别是与能量定理和庞加莱多项式的关系。

提出的方法

• 在带符号的有限单图范畴上定义三种交换环结构：弱积（笛卡尔积）、直积（张量积）和强积（组合积），其中不相交并集作为加法运算。
• 使用格罗滕迪克群构造法，将图在不相交并集下的幺半群扩展为群，从而允许形式减法与环结构。
• 应用图补运算将齐科夫环对偶化为强环，建立两者之间的同构关系。
• 证明强积同伦于斯坦利-雷伊纳尔德积，从而可应用代数拓扑工具。
• 应用库恩尼茨公式，证明庞加莱多项式在强积下具有乘法性，从而确认环兼容性。
• 使用 Mathematica 编写的计算代码验证：强积的连接拉普拉斯算子等于各自主连接拉普拉斯算子的张量积，且能量等于欧拉示性数。

实验结果

研究问题

• RQ1 图的强环是否同构于已知代数结构（如斯坦利-雷伊纳尔德环）？
• RQ2 欧拉示性数在图的强积下是否保持环同态性质？
• RQ3 团数能否扩展为从齐科夫环到整数的环同态？
• RQ4 连接拉普拉斯算子的谱在图的强积下如何表现？
• RQ5 图的能量（定义为连接拉普拉斯算子逆矩阵元素之和）在强积下是否具有乘法性？

主要发现

• 图的强环在同伦等价下同构于斯坦利-雷伊纳尔德环，确认庞加莱多项式满足库恩尼茨公式。
• 欧拉示性数是从强环到整数的环同态，满足 χ(G × H) = χ(G)χ(H) 与 χ(G + H) = χ(G) + χ(H)。
• 团数（定义为 dim(G) + 1）是从齐科夫环到 ℤ 的环同态，保持乘法结构。
• 两个图的强积的连接拉普拉斯算子等于其各自连接拉普拉斯算子的张量积，确认谱兼容性。
• 强积的连接拉普拉斯算子谱满足 σ(L(G × H)) = σ(L(G)) × σ(L(H))，且能量（逆矩阵元素之和）等于欧拉示性数的乘积。
• 强积同时包含弱积与直积作为子图，且环结构与同伦、上同调及谱理论兼容。