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Large vertex-symmetric Higman graphs with $\mu=6$
N. D. Zyulyarkinaa, M. Kh. Shermetovab a South Ural State University, Chelyabinsk
b Kabardino-Balkar State University, Nal'chik
Abstract:
A strongly regular graph with $v={m\choose 2}$ and $k=2(m-2)$ is called a Higman graph. In such a graph, the parameter $\mu$ is 4, 6, 7, or 8. If $\mu=6$, then $m\in\{9,17,27,57\}$. Vertex-symmetric Higman graphs were classified by N.D. Zyulyarkina and A.A. Makhnev (all of these graphs turned out to have rank 3). A program of classification of vertex-symmetric Higman graphs is implemented. Earlier Zyulyarkina and Makhnev found vertex-symmetric Higman graphs with $\mu=6$ and $m\in\{9,17\}$. In the present paper, vertex-symmetric Higman graphs with $\mu=6$ and $m\in{27,57}$ are studied. It is interesting that the group $G/S(G)$ may contain two components $L$ and $M$. In the case $m=27$, we have $M\cong A_5,A_6$ and $L\cong L_3(3)$; in the case $m=57$, we have either $M\cong PSp_4(3)$ and $L\cong L_3(7)$ or $M\cong A_6$ and $L\cong J_1$.
Keywords:
distance-regular graph, graph automorphism.
Received: 20.02.2018 Revised: 16.10.2018 Accepted: 22.10.2018
Citation:
N. D. Zyulyarkina, M. Kh. Shermetova, “Large vertex-symmetric Higman graphs with $\mu=6$”, Trudy Inst. Mat. i Mekh. UrO RAN, 24, no. 4, 2018, 146–155
Linking options:
https://www.mathnet.ru/eng/timm1582 https://www.mathnet.ru/eng/timm/v24/i4/p146
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Abstract page: | 126 | Full-text PDF : | 38 | References: | 37 | First page: | 1 |
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