Geodesic
On the eigenvalues multiplicity function of the Star graph
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 15 (2018), pp. 1416-1425.

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

The Star graph is the Cayley graph on the symmetric group $\mathrm{Sym}_n$ generated by the set of transpositions $\{(1 2),(1 3),\ldots,(1 n)\}$. We consider the spectrum of the Star graph as the spectrum of its adjacency matrix. The spectrum of $S_n$ is integral as it was shown independently by R. Krakovski, B. Mohar, and G. Chapuy, V. Feray in 2012. In this paper we show that the multiplicity of eigenvalues of the Star graph is a polynomial in the indeterminate $n$ of degree $2(t-1)$ with leading coefficient $\frac{1}{(t-1)!}$.
Keywords: Cayley graph, Star graph, symmetric group, graph spectrum, eigenvalues; multiplicity. 
@article{SEMR_2018_15_a78,
     author = {E. N. Khomyakova},
     title = {On the eigenvalues multiplicity function of the {Star} graph},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1416--1425},
     publisher = {mathdoc},
     volume = {15},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2018_15_a78/}
}
TY  - JOUR
AU  - E. N. Khomyakova
TI  - On the eigenvalues multiplicity function of the Star graph
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2018
SP  - 1416
EP  - 1425
VL  - 15
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2018_15_a78/
LA  - ru
ID  - SEMR_2018_15_a78
ER  - 
%0 Journal Article
%A E. N. Khomyakova
%T On the eigenvalues multiplicity function of the Star graph
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2018
%P 1416-1425
%V 15
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2018_15_a78/
%G ru
%F SEMR_2018_15_a78
E. N. Khomyakova. On the eigenvalues multiplicity function of the Star graph. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 15 (2018), pp. 1416-1425. http://geodesic.mathdoc.fr/item/SEMR_2018_15_a78/

[1] N. Alon, O. Ahmadi, I. F. Blake, I. E. Shparlinski, “Graphs with integral spectrum”, Linear Algebra and its Applications, 430 (2009), 547–552 | DOI | MR | Zbl

[2] S. Avgustinovich, E. Khomyakova, E. Konstantinova, “Multiplicities of eigenvalues of the Star graph”, Sib. Electron. Math. Reports, 13 (2016), 1258–1270 | MR | Zbl

[3] G. Chapuy, V. Feray, A note on a Cayley graph of $Sym_{n}$, 2012, 3 pp., arXiv: 1202.4976v2

[4] J.S. Frame, G. de B. Robinson, R.M. Thrall, “The hook graphs of the symmetric group”, Canad. J. Math., 6 (1954), 316–325 | DOI | MR | Zbl

[5] E. N. Khomyakova, E. V. Konstantinova, “Note on exact values of multiplicities of eigenvalues of the Star graph”, Sib. Electron. Math. Reports, 12 (2015), 92–100 | MR | Zbl

[6] E. V. Konstantinova, Some problems on Cayley graphs, University of Primorska Press, Koper, 2013, 98 pp. http://www.hippocampus.si/ISBN/978-961-6832-51-9/index.html

[7] R. Krakovski, B. Mohar, “Spectrum of Cayley Graphs on the Symmetric Group generated by transposition”, Linear Algebra and its applications, 437 (2012), 1033–1039 | DOI | MR | Zbl

[8] B. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, second edition, Springer, New York, 2001 | MR | Zbl