图与完全图的结构等价性
Structural Equivalence in Graphs and Complete Skeletons
摘要：
图$Γ$的两个顶点$u$和$v$是结构等价的当且仅当转置$(u\,v)$在$Γ$的自同构群Aut（$Γ$）中.讨论了Aut（$Γ$）中的结构等价和由转置生成的顶点置换群的一些性质，以及这些置换群的素图.结构等价的概念被用来开发一种将图重新配置为所谓的完整骨架的方法，这与压缩图密切相关。最后，利用图$Γ$的完全骨架（记为$Ω(Γ)$），给出了图的秩$(I+A(Γ))$的计算公式，该公式有助于确定图$Γ$的-1特征值的重数.
英文摘要：
Two vertices $u$ and $v$ of a graph $Γ$ are strucuturally equivalent if and only if the transposition $(u\,v)$ is in Aut($Γ$), the automorphism group of $Γ$. Some properties of structural equivalence and the group of vertex permutations generated by the transpositions in Aut($Γ$) are discussed, along with the prime graphs of these groups. The notion of structural equivalence is used to develop a way of reconfiguring graphs into what are called their complete skeletons, which is closely related to compression graphs. Finally, the complete skeleton of a graph $Γ$, denoted $Ω(Γ)$, is used to find a formula for rank$(I+A(Γ))$, which is helpful for determining the multiplicity of the -1 eigenvalue of $Γ$.
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。
