关于身份Seidel开关
On identity Seidel switches
AI总结：
本文研究了保持图不变的Seidel开关，揭示了其结构特性及度数参数的约束条件，并提出相关构造与开放问题。
摘要：
Seidel开关是图上的一种经典操作，在两个图理论、符号图和开关类理论中扮演着核心角色。本文聚焦于那些在同构意义下保持给定图不变的开关。我们称这样的顶点子集为"身份Seidel开关"。在回顾Seidel开关的基本性质及其相关的阿贝尔群结构后，我们引入图的Seidel等价类，然后研究固定图的身份Seidel开关家族的结构。我们证明该家族在复合下形成一个2-群，并获得对图的结构约束，其中许多顶点或边会产生身份开关。特别是，我们通过度数参数导出必要条件，并通过诱导子图的自同构体来刻画某些边身份开关。一些构造和例子被提出，一些开放问题也被提出。
英文摘要：
An identity Seidel switch of a graph is a vertex subset whose Seidel switch produces an isomorphic graph. We distinguish the elementary abelian group of switching operations from the family of identity switches of a fixed graph, which need not be a group. We obtain an edge-count condition for identity switches and deduce that every vertex-identity switch has degree half the number of remaining vertices. In particular, no graph of order at least two has every vertex as an identity switch. For a two-vertex switching set inducing an edge, we characterize the switches admitting an isomorphism that preserves the endpoint pair, in terms of automorphisms of the remaining induced subgraph. We establish modification results under this additional hypothesis and give explicit constructions of identity switches. Counterexamples show why edge-count conditions and unrestricted modification assertions do not suffice.
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。
