The prime spectrum of an automorphism group of an 𝐴𝑇4(𝑝,𝑝+2,𝑟)-graph

Tsiovkina, L.

doi:10.1090/spmj/1677

The prime spectrum of an automorphism group of an $\mathrm {AT4}(p,p+2,r)$-graph
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by L. Yu. Tsiovkina
Translated by: the author

St. Petersburg Math. J. 32 (2021), 917-928

DOI: https://doi.org/10.1090/spmj/1677

Published electronically: August 31, 2021

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Abstract:

The present paper is devoted classification of $\mathrm {AT4}(p,p+2,r)$-graphs. There is a unique $\mathrm {AT4}(p,p+2,r)$-graph with $p=2$, namely, the distance-transitive Soicher graph with intersection array $\{56, 45, 16, 1;1, 8, 45, 56\}$, whose local graphs are isomorphic to the Gewirtz graph. The existence of an $\mathrm {AT4}(p,p+2,r)$-graph with ${p>2}$ remains an open question. It is known that the local graphs of each $\mathrm {AT4}(p,p+2,r)$-graph are strongly regular with parameters $\big ((p+2)(p^2+4p+2),p(p+3),p-2,p\big )$. In this paper, an upper bound is found for the prime spectrum of the automorphism group of a strongly regular graph with such parameters, and also some restrictions obtained for the prime spectrum and the structure of the automorphism group of an $\mathrm {AT4}(p,p+2,r)$-graph in the case when $2<p$ is a prime power. As a corollary, it is shown that there are no arc-transitive $\mathrm {AT}(p,p+2,r)$-graphs with $p\in \{11,17,27\}$.

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