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On graphs and codes preserved by edge local complementation

机译：关于通过边缘局部互补保留的图和代码

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摘要

Orbits of graphs under local complementation (LC) and edge local complementation (ELC) have been studied in several different contexts. For instance, there are connections between orbits of graphs and error-correcting codes. We define a new graph class, ELC-preserved graphs, comprising all graphs that have an ELC orbit of size one. Through an exhaustive search, we find all ELC-preserved graphs of order up to 12 and all ELC-preserved bipartite graphs of order up to 16. We provide general recursive constructions for infinite families of ELC-preserved graphs, and show that all known ELC-preserved graphs arise from these constructions or can be obtained from Hamming codes. We also prove that certain pairs of ELC-preserved graphs are LC equivalent. We define ELC-preserved codes as binary linear codes corresponding to bipartite ELC-preserved graphs, and study the parameters of such codes.
机译：在几种不同的情况下研究了图的局部补全（LC）和边缘局部补全（ELC）下的轨道。例如，图的轨道与纠错码之间存在联系。我们定义一个新的图类，保留ELC的图，包括所有ELC轨道为1的图。通过详尽的搜索，我们找到了所有不超过12阶的ELC保留图和不超过16阶的所有ELC保留二部图。我们为无限个ELC保留图族提供了通用的递归构造，并证明了所有已知的ELC保留的图由这些结构产生或可以从汉明码获得。我们还证明某些ELC保留图对等效于LC。我们将保留ELC的代码定义为对应于二分ELC保留图的二进制线性代码，并研究此类代码的参数。

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