On automorphisms of a distance-regular graph with intersection array $\{125,96,1;1,48,125\}$
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 159 (2017) no. 1, pp. 13-20
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J. Koolen posed the problem of studying distance-regular graphs in which neighborhoods of vertices are strongly regular graphs with the second eigenvalue $\leq t$ for the given positive integer $t$. This problem is reduced to the description of distance-regular graphs in which neighborhoods of vertices are strongly regular graphs with the non-principal eigenvalue $t$ for $t =1,2,\dots$ In the paper “Distance regular graphs in which local subgraphs are strongly regular graphs with the second eigenvalue at most 3”, A. A. Makhnev and D. V. Paduchikh found the arrays of intersections of distance-regular graphs in which neighborhoods of vertices are strongly regular graphs with the second eigenvalue t such as $2$. The graphs with intersection arrays $\{125,96,1;1,48,125\}$, $\{176,150,1; 1,25,176\}$, and $\{256,204,1;1,51,256\}$ remain unexplored. In this paper, we have found the possible orders and the structures of subgraphs of the fixed points of automorphisms of a distance-regular graph with the intersection array $\{125,96,1;1,48,125\}$. It has been proved that the neighborhoods of the vertices of this graph are pseudogeometric graphs for $GQ(4,6)$. Composition factors of the automorphism group of a distance-regular graph with the intersection array $\{125.96,1;1,48,125\}$ have been determined.
Keywords:
distance-regular graph, automorphism groups of graph.
@article{UZKU_2017_159_1_a1,
author = {V. V. Bitkina and A. A. Makhnev},
title = {On automorphisms of a~distance-regular graph with intersection array $\{125,96,1;1,48,125\}$},
journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
pages = {13--20},
publisher = {mathdoc},
volume = {159},
number = {1},
year = {2017},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/UZKU_2017_159_1_a1/}
}
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AU - A. A. Makhnev
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V. V. Bitkina; A. A. Makhnev. On automorphisms of a distance-regular graph with intersection array $\{125,96,1;1,48,125\}$. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 159 (2017) no. 1, pp. 13-20. http://geodesic.mathdoc.fr/item/UZKU_2017_159_1_a1/