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PENTAVALENT SYMMETRIC GRAPHS OF ORDER $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}30p$

Part of: Graph theory

Published online by Cambridge University Press:  27 August 2014

BO LING ,
CI XUAN WU  and
BEN GONG LOU
BO LING
Affiliation:
School of Mathematics and Statistics, Yunnan University, Kunmin 650031, PR China email [email protected]
CI XUAN WU
Affiliation:
School of Mathematics and Statistics, Yunnan University, Kunmin 650031, PR China email [email protected]
BEN GONG LOU*
Affiliation:
School of Mathematics and Statistics, Yunnan University, Kunmin 650031, PR China email [email protected]
*
✉[email protected]
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Abstract

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A complete classification is given of pentavalent symmetric graphs of order $30p$, where $p\ge 5$ is a prime. It is proved that such a graph ${\Gamma }$ exists if and only if $p=13$ and, up to isomorphism, there is only one such graph. Furthermore, ${\Gamma }$ is isomorphic to $\mathcal{C}_{390}$, a coset graph of PSL(2, 25) with ${\sf Aut}\, {\Gamma }=\mbox{PSL(2, 25)}$, and ${\Gamma }$ is 2-regular. The classification involves a new 2-regular pentavalent graph construction with square-free order.

Keywords

arc-transitive graphpentavalent symmetric graphautomorphism group

MSC classification

Secondary: 05C30: Enumeration in graph theory 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 3 , December 2014 , pp. 353 - 362
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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