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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
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 90 / Issue 3 / December 2014
- Published online by Cambridge University Press:
- 27 August 2014, pp. 353-362
- Print publication:
- December 2014
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- Article
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- You have access
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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.
