Geodesic
Large vertex-symmetric Higman graphs with $\mu=6$
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 4, pp. 146-155
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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
Mots-clés : graph automorphism. 
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N. D. Zyulyarkina; M. Kh. Shermetova. Large vertex-symmetric Higman graphs with $\mu=6$. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 4, pp. 146-155. http://geodesic.mathdoc.fr/item/TIMM_2018_24_4_a11/

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