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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Analysing finite locally $s$-arc transitive graphs


Authors: Michael Giudici, Cai Heng Li and Cheryl E. Praeger
Journal: Trans. Amer. Math. Soc. 356 (2004), 291-317
MSC (2000): Primary 05C25, 20B25
Published electronically: August 25, 2003
MathSciNet review: 2020034
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Abstract: We present a new approach to analysing finite graphs which admit a vertex intransitive group of automorphisms $G$ and are either locally $(G,s)$-arc transitive for $s \geq 2$ or $G$-locally primitive. Such graphs are bipartite with the two parts of the bipartition being the orbits of $G$. Given a normal subgroup $N$ which is intransitive on both parts of the bipartition, we show that taking quotients with respect to the orbits of $N$ preserves both local primitivity and local $s$-arc transitivity and leads us to study graphs where $G$ acts faithfully on both orbits and quasiprimitively on at least one. We determine the possible quasiprimitive types for $G$ in these two cases and give new constructions of examples for each possible type. The analysis raises several open problems which are discussed in the final section.


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Additional Information

Michael Giudici
Affiliation: School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia
Email: giudici@maths.uwa.edu.au

Cai Heng Li
Affiliation: School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia
Email: li@maths.uwa.edu.au

Cheryl E. Praeger
Affiliation: School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia
Email: praeger@maths.uwa.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03361-0
PII: S 0002-9947(03)03361-0
Received by editor(s): November 22, 2002
Published electronically: August 25, 2003
Additional Notes: This paper forms part of an Australian Research Council large grant project which supported the first author. The second author was supported by an ARC Fellowship
Article copyright: © Copyright 2003 American Mathematical Society