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On Graphs in Which the Neighborhoods of Vertices Are Edge-Regular Graphs without 3-Claws. / Chen, M.; Makhnev, A.; Nirova, M.
In: Proceedings of the Steklov Institute of Mathematics, Vol. 323, No. S1, 01.12.2023, p. S53-S55.

Research output: Contribution to journal › Article › peer-review

Harvard

Chen, M, Makhnev, A & Nirova, M 2023, 'On Graphs in Which the Neighborhoods of Vertices Are Edge-Regular Graphs without 3-Claws', Proceedings of the Steklov Institute of Mathematics, vol. 323, no. S1, pp. S53-S55. https://doi.org/10.1134/S0081543823060044

APA

Chen, M., Makhnev, A., & Nirova, M. (2023). On Graphs in Which the Neighborhoods of Vertices Are Edge-Regular Graphs without 3-Claws. Proceedings of the Steklov Institute of Mathematics, 323(S1), S53-S55. https://doi.org/10.1134/S0081543823060044

Vancouver

Chen M, Makhnev A, Nirova M. On Graphs in Which the Neighborhoods of Vertices Are Edge-Regular Graphs without 3-Claws. Proceedings of the Steklov Institute of Mathematics. 2023 Dec 1;323(S1):S53-S55. doi: 10.1134/S0081543823060044

Author

Chen, M. ; Makhnev, A. ; Nirova, M. / On Graphs in Which the Neighborhoods of Vertices Are Edge-Regular Graphs without 3-Claws. In: Proceedings of the Steklov Institute of Mathematics. 2023 ; Vol. 323, No. S1. pp. S53-S55.

BibTeX

@article{b54033319eef4d9b848bc375c0a4ed97,
title = "On Graphs in Which the Neighborhoods of Vertices Are Edge-Regular Graphs without 3-Claws",
abstract = "The triangle-free Krein graph Kre(r) is strongly regular with parameters (Formula presented.). 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 Kre3 does not exist. Later Makhnev proved that the graph Kre4 does not exist. The graph Kre(r) is the only strongly regular triangle-free graph in which the antineighborhood of a vertex (Formula presented.) is strongly regular. The graph (Formula presented.) has parameters (Formula presented.). This work clarifies Makhnev{\textquoteright}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 (Formula presented.) exists and is the complement of the block graph of a quasi-symmetric 2-design.",
author = "M. Chen and A. Makhnev and M. Nirova",
note = "This research was supported by the National Natural Science Foundation of China (project no. 12171126) and by a grant from the Engineering Modeling and Statistical Computing Laboratory of the Hainan Province.",
year = "2023",
month = dec,
day = "1",
doi = "10.1134/S0081543823060044",
language = "English",
volume = "323",
pages = "S53--S55",
journal = "Proceedings of the Steklov Institute of Mathematics",
issn = "0081-5438",
publisher = "Pleiades Publishing",
number = "S1",

}

RIS

TY - JOUR

T1 - On Graphs in Which the Neighborhoods of Vertices Are Edge-Regular Graphs without 3-Claws

AU - Chen, M.

AU - Makhnev, A.

AU - Nirova, M.

N1 - This research was supported by the National Natural Science Foundation of China (project no. 12171126) and by a grant from the Engineering Modeling and Statistical Computing Laboratory of the Hainan Province.

PY - 2023/12/1

Y1 - 2023/12/1

N2 - The triangle-free Krein graph Kre(r) is strongly regular with parameters (Formula presented.). 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 Kre3 does not exist. Later Makhnev proved that the graph Kre4 does not exist. The graph Kre(r) is the only strongly regular triangle-free graph in which the antineighborhood of a vertex (Formula presented.) is strongly regular. The graph (Formula presented.) has parameters (Formula presented.). 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 (Formula presented.) exists and is the complement of the block graph of a quasi-symmetric 2-design.

AB - The triangle-free Krein graph Kre(r) is strongly regular with parameters (Formula presented.). 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 Kre3 does not exist. Later Makhnev proved that the graph Kre4 does not exist. The graph Kre(r) is the only strongly regular triangle-free graph in which the antineighborhood of a vertex (Formula presented.) is strongly regular. The graph (Formula presented.) has parameters (Formula presented.). 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 (Formula presented.) exists and is the complement of the block graph of a quasi-symmetric 2-design.

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U2 - 10.1134/S0081543823060044

DO - 10.1134/S0081543823060044

M3 - Article

VL - 323

SP - S53-S55

JO - Proceedings of the Steklov Institute of Mathematics

JF - Proceedings of the Steklov Institute of Mathematics

SN - 0081-5438

IS - S1

ER -

ID: 53754191