同构的推广及其应用
A Generalisation of Isomorphisms with Applications
摘要：
在本文中，我们研究的TF-同构，同构的自然推广的行为。TF-同构允许我们简化处理看似无关问题的方法。特别是，我们提到的邻域重建问题，矩阵对称化问题和图的稳定性。我们首先研究TF-同构下的不变性。特别是，我们表明，交替的踪迹和发病率双覆盖的TF-同构是保守的，无论他们是TF-同构之间的图或有向图。然后，我们定义了一个等价关系，并随后将其等价类关联到图的关联双覆盖。通过将关联双覆盖的边从一个颜色类引导到另一个颜色类并丢弃孤立顶点，我们得到了TF-同构下的不变量，它聚集了许多不变量。这可以用来研究TF-轨道，一个类似的推广的轨道的置换群。
英文摘要：
In this paper, we study the behaviour of TF-isomorphisms, a natural generalisation of isomorphisms. TF-isomorphisms allow us to simplify the approach to seemingly unrelated problems. In particular, we mention the Neighbourhood Reconstruction problem, the Matrix Symmetrization problem and Stability of Graphs. We start with a study of invariance under TF-isomorphisms. In particular, we show that alternating trails and incidence double covers are conserved by TF-isomorphisms, irrespective of whether they are TF-isomorphisms between graphs or digraphs. We then define an equivalence relation and subsequently relate its equivalence classes to the incidence double cover of a graph. By directing the edges of an incidence double cover from one colour class to the other and discarding isolated vertices we obtain an invariant under TF-isomorphisms which gathers a number of invariants. This can be used to study TF-orbitals, an analogous generalisation of the orbitals of a permutation group.
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。
