图的代数不变式;基于计算机探索的研究
Algebraic invariants of graphs; a study based on computer exploration
摘要：
本文考虑n阶加权图上多项式不变量环I_n。我们的主要兴趣是使用这个环来定义和探索图的同构问题的代数版本，如乌拉姆的重建猜想。有大量的文献不变量理论提供了一般的结果和算法。然而，在所涉及的计算中存在组合爆炸，并且据我们所知，环I_n仅在n<=4时被完全描述。这使我们研究环I_n本身。我们使用密集的计算机探索小n，并开发PerMuVAR，MuPAD的库，用于计算置换群的不变环。本文给出环I_n的一般性质，以及对小n的计算机探索结果，包括构造I_5的一个中等生成集。我们讨论了由这些结果（低次参数系，单峰性）所提出的几个结构，对于I_n以及更一般的不变环.我们还表明，一些特定的集不生成，反驳了一个猜想的Pouzet有关的重建，以及一个引理的Grigoriev不变环有向图。最后，我们提供了一个非常简单的不变量域的最小生成集。
英文摘要：
We consider the ring I_n of polynomial invariants over weighted graphs on n vertices. Our primary interest is the use of this ring to define and explore algebraic versions of isomorphism problems of graphs, such as Ulam's reconstruction conjecture. There is a huge body of literature on invariant theory which provides both general results and algorithms. However, there is a combinatorial explosion in the computations involved and, to our knowledge, the ring I_n has only been completely described for n<=4. This led us to study the ring I_n in its own right. We used intensive computer exploration for small n, and developed PerMuVAR, a library for MuPAD, for computing in invariant rings of permutation groups. We present general properties of the ring I_n, as well as results obtained by computer exploration for small n, including the construction of a medium sized generating set for I_5. We address several conjectures suggested by those results (low degree system of parameters, unimodality), for I_n as well as for more general invariant rings. We also show that some particular sets are not generating, disproving a conjecture of Pouzet related to reconstruction, as well as a lemma of Grigoriev on the invariant ring over digraphs. We finally provide a very simple minimal generating set of the field of invariants.
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