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Sib. Èlektron. Mat. Izv., 2016, Volume 13, Pages 286–299 (Mi semr672)  

Discrete mathematics and mathematical cybernetics

The number of small cycles in the Star graph

Alexey N. Medvedevabc

a Sobolev Institute of Mathematics, 4, Koptyug av., 630090, Novosibirsk, Russia
b MTA Rényi Alfréd Institute of Mathematics, Reáltanoda ut. 13-15., Budapest, 1053, Hungary
c Central European University, Nador ut. 9, Budapest, 1051, Hungary

Abstract: 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.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-01-05867_а
Supported by Grant 15–01–05867 of the Russian Foundation of Basic Research.


DOI: https://doi.org/10.17377/semi.2016.13.023

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Document Type: Article
UDC: 519.1
MSC: 05C25
Received January 21, 2016, published April 29, 2016
Language: English

Citation: Alexey N. Medvedev, “The number of small cycles in the Star graph”, Sib. Èlektron. Mat. Izv., 13 (2016), 286–299

Citation in format AMSBIB
\Bibitem{Med16}
\by Alexey~N.~Medvedev
\paper The number of small cycles in the Star graph
\jour Sib. \`Elektron. Mat. Izv.
\yr 2016
\vol 13
\pages 286--299
\mathnet{http://mi.mathnet.ru/semr672}
\crossref{https://doi.org/10.17377/semi.2016.13.023}


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