A graph with intersection array {18, 15, 1; 1, 5, 18} is not vertex-symmetric

K. S. Efimov<sup>abc

a</sup> Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg
^b Ural State University of Economics, Ekaterinburg
^c Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg

Abstract: A.A. Makhnev and V.P. Burichenko found possible intersection arrays of distance-regular locally cyclic graphs with at most 1000 vertices. They proposed a program for studying arc-transitive graphs with these intersection arrays. The neighborhood of a vertex in such a graph is the union of isolated polygons. We study automorphisms of a hypothetical distance-regular graph with intersection array {18, 15, 1; 1, 5, 18}. In particular, we prove that the automorphism group of this graph acts intransitively on the vertex set.

Keywords: distance-regular graph, graph automorphism.

UDC: 519.17

MSC: 05C25, 20B25

Received: 26.06.2018

DOI: 10.21538/0134-4889-2018-24-3-62-67

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