The prime spectrum of an automorphism group of an $\mathrm {AT4}(p,p+2,r)$-graph
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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 $p\in \{11,17,27\}$.References
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Bibliographic Information
- L. Yu. Tsiovkina
- Affiliation: N. N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi, 16, Yekaterinburg 620108, Russia
- Email: l.tsiovkina@gmail.com
- Received by editor(s): February 27, 2019
- Published electronically: August 31, 2021
- © Copyright 2021 American Mathematical Society
- Journal: St. Petersburg Math. J. 32 (2021), 917-928
- MSC (2020): Primary 05E18; Secondary 05E30, 05C25
- DOI: https://doi.org/10.1090/spmj/1677
- MathSciNet review: 4167876