On Graphs in Which the Neighborhoods of Vertices Are Edge-Regular Graphs without 3-Claws

Pub Date : 2023-12-01 DOI:10.1134/s0081543823060044

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Abstract

The triangle-free Krein graph Kre \((r)\) is strongly regular with parameters \(((r^{2}+3r)^{2},\) \(r^{3}+3r^{2}+r,0,r^{2}+r)\) . The existence of such graphs is known only for \(r=1\) (the complement of the Clebsch graph) and \(r=2\) (the Higman–Sims graph). A.L. Gavrilyuk and A.A. Makhnev proved that the graph Kre \((3)\) does not exist. Later Makhnev proved that the graph Kre \((4)\) does not exist. The graph Kre \((r)\) is the only strongly regular triangle-free graph in which the antineighborhood of a vertex Kre \((r)^{\prime}\) is strongly regular. The graph Kre \((r)^{\prime}\) has parameters \(((r^{2}+2r-1)(r^{2}+3r+1),r^{3}+2r^{2},0,r^{2})\) . This work clarifies Makhnev’s result on graphs in which the neighborhoods of vertices are strongly regular graphs without \(3\) -cocliques. As a consequence, it is proved that the graph Kre \((r)\) exists if and only if the graph Kre \((r)^{\prime}\) exists and is the complement of the block graph of a quasi-symmetric \(2\) -design.

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论顶点邻域为无三爪边缘规则图的图

Abstract The triangle-free Krein graph Kre \((r)\) is strongly regular with parameters \(((r^{2}+3r)^{2},\) . \(r^{3}+3r^{2}+r,0,r^{2}+r)\) .我们只知道 \(r=1\)（克莱布什图的补集）和 \(r=2\)（希格曼-西姆斯图）存在这样的图。A.L. Gavrilyuk 和 A.A. Makhnev 证明了 Kre\((3)\) 图并不存在。后来马赫涅夫又证明了 Kre （(4)）图不存在。图 Kre （(r)）是唯一的强正则无三角形图，其中顶点 Kre （(r)^{\prime}\）的反邻域是强正则的。图 Kre \((r)^{\prime}\) 的参数是 \(((r^{2}+2r-1)(r^{2}+3r+1),r^{3}+2r^{2},0,r^{2})\)。这项工作澄清了马赫涅夫关于顶点邻域为不含\(3\) -楔的强规则图的结果。结果证明，当且仅当图 Kre \((r)^{\prime}\) 存在并且是准对称 \(2\) -设计的块图的补集时，图 Kre \((r)\) 才存在。

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