Geodesic
The number of small cycles in the Star graph
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 13 (2016), pp. 286-299

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The Star graph is the Cayley graph on the symmetric group $Sym_n$ generated by the set of transpositions $\{(1\,i) \in Sym_n: 2 \leqslant i \leqslant n\}$. This graph is bipartite and does not contain odd cycles but contains all even cycles with a sole exception of $4$-cycles. We denote as $(\pi,id)$-cycles the cycles constructed from two shortest paths between a given vertex $\pi$ and the identity $id$. In this paper we derive the exact number of $(\pi,id)$-cycles for particular structures of the vertex $\pi$. We use these results to obtain the total number of $10$-cycles passing through any given vertex in the Star graph.
Keywords: Cayley graphs; Star graph; cycle embedding; number of cycles. 
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     author = {Alexey N. Medvedev},
     title = {The number of small cycles in the {Star} graph},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {286--299},
     publisher = {mathdoc},
     volume = {13},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2016_13_a57/}
}
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Alexey N. Medvedev. The number of small cycles in the Star graph. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 13 (2016), pp. 286-299. http://geodesic.mathdoc.fr/item/SEMR_2016_13_a57/