Geodesic
Perfect codes from $\mathrm{PGL}(2,5)$ in Star graphs
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 534-539.

Voir la notice de l'article provenant de la source Math-Net.Ru

The Star graph $S_n$ is the Cayley graph of the symmetric group $\mathrm{Sym}_n$ with the generating set $\{(1\ i): 2\leq i\leq n \}$. Arumugam and Kala proved that $\{\pi\in \mathrm{Sym}_n: \pi(1)=1\}$ is a perfect code in $S_n$ for any $n$, $n\geq 3$. In this note we show that for any $n$, $n\geq 6$ the Star graph $S_n$ contains a perfect code which is the union of cosets of the embedding of $\mathrm{PGL}(2,5)$ into $\mathrm{Sym}_6$.
Keywords: perfect code, efficient dominating set, Cayley graph, Star graph, projective linear group, symmetric group. 
@article{SEMR_2020_17_a67,
     author = {I. Yu. Mogilnykh},
     title = {Perfect codes from $\mathrm{PGL}(2,5)$ in {Star} graphs},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {534--539},
     publisher = {mathdoc},
     volume = {17},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2020_17_a67/}
}
TY  - JOUR
AU  - I. Yu. Mogilnykh
TI  - Perfect codes from $\mathrm{PGL}(2,5)$ in Star graphs
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2020
SP  - 534
EP  - 539
VL  - 17
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2020_17_a67/
LA  - en
ID  - SEMR_2020_17_a67
ER  - 
%0 Journal Article
%A I. Yu. Mogilnykh
%T Perfect codes from $\mathrm{PGL}(2,5)$ in Star graphs
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2020
%P 534-539
%V 17
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2020_17_a67/
%G en
%F SEMR_2020_17_a67
I. Yu. Mogilnykh. Perfect codes from $\mathrm{PGL}(2,5)$ in Star graphs. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 534-539. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a67/

[1] S. Akers, B. Krishnamurthy, “A group theoretic model for symmetric interconnection networks”, Proceedings of the International Conference on Parallel Processing, 1986, 216–223 | Zbl

[2] S. V. Avgustinovich, E. N. Khomyakova, E. V. Konstantinova, “Multiplicities of eigenvalues of the Star graph”, Siberian Electronic Mathematical Reports, 13 (2016), 1258–1270 | MR | Zbl

[3] S. Arumugam, R. Kala, “Domination parameters of star graphs”, Ars. Combin., 44 (1996), 93–96 | MR | Zbl

[4] J. D. Dixon, B. Mortimer, Permutation Groups, Springer-Verlag, New York–Berlin–Heidelberg, 1996 | MR | Zbl

[5] S. Goryainov, V. V. Kabanov, E. Konstantinova, L. Shalaginov, A. Valyuzhenich, “PI-eigenfunctions of the Star graphs”, Linear Algebra and its Applications, 586 (2020), 7–27 | DOI | MR | Zbl

[6] V. V. Kabanov, E. Konstantinova, L. Shalaginov, A. Valyuzhenich, Minimum supports of eigenfunctions with the second largest eigenvalue of the Star graph, October 2019, arXiv: 1910.01374 | MR

[7] W. Guo, D. V. Lytkina, V. D. Mazurov, D. O. Revin, “Spectra of Cayley graphs”, Algebra and Logic, 58 (2019), 297–305 | DOI | Zbl

[8] G. Chapuy, V. Feray, A note on a Cayley graph of Sym(n), February 2012, arXiv: 1202.4976v2

[9] Y.-Q Feng, “Automorphism groups of Cayley graphs on symmetric groups with generating transposition sets”, J. Combin. Theory Ser. B, 96 (2006), 67–72 | DOI | MR | Zbl

[10] D. S. Krotov, “Trades in the combinatorial configurations”, XII International Seminar Discrete Mathematics and its Applications (Moscow, 20–25 June, 2016), 84–96 (in Russian)

[11] T. Etzion, S. Buzaglo, “Bounds on the Size of Permutation Codes With the Kendall tau-Metric”, IEEE Trans. Inform. Theory, 61:6 (2015), 3241–3250 | DOI | MR | Zbl

[12] I. J. Dejter, O. Tomaiconza, “Nonexistence of efficient dominating sets in the Cayley graphs generated by transposition trees of diameter 3”, Discrete Applied Mathematics, 232 (2017), 116–124 | DOI | MR | Zbl

[13] A. Jiang, R. Mateescu, M. Schwartz, J. Bruck, “Rank modulation for flash memories”, IEEE Trans. on Inform. Theory, 55 (2009), 2659–2673 | DOI | MR | Zbl

[14] Yu. L. Vasil'ev, “On Nongroup Closely Packed Codes”, Probl. Kibern., 8 (1962), 337–339 | MR | Zbl