We prove that any completely regular code with minimum eigenvalue in any geometric graph G corresponds to a completely regular code in the clique graph of G. Studying the interrelation of these codes, a complete characterization of the completely regular codes in the Johnson graphs J(n,w) with covering radius w-1 and strength 1 is obtained. In particular this result finishes a characterization of the completely regular codes in the Johnson graphs J(n,3). We also classify the completely regular codes of strength 1 in the Johnson graphs J(n,4) with only one case for the eigenvalues left open.
翻译：我们证明,在任何几何图G中,任何完全正常的代码,其最小值为igen值,与G级分类图中的完全正常代码相对应。研究这些代码的相互关系,获得了约翰逊图J(n,w)中完全正常代码的完整特征,覆盖半径W-1和强度1。特别是,这一结果完成了约翰逊图J(n,3)中对完全正常代码1的定性。我们还对约翰逊图J(n,4)中完全正常的强度代码1进行了分类,只有1个案例涉及egen值。
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