Geodesic
Multiplicities of eigenvalues of the Star graph
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 13 (2016), pp. 1258-1270

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The Star graph $S_n$, $n\geqslant 2$, is the Cayley graph on the symmetric group $\mathrm{Sym}_n$ generated by the set of transpositions [4] $\{(1~2), (1~3), \ldots, (1~n)\}$. We consider the spectrum of the Star graph as the spectrum of its adjacency matrix. It is known that the spectrum of $S_n$ is integral. Analytic formulas for multiplicities of eigenvalues $\pm(n-k)$ for $k = 2, 3, 4, 5$ in the Star graph are given in this paper. We also prove that any fixed integer has multiplicity at least $2^{\frac{1}{2}n \log n (1-o(1))}$ as an eigenvalue of $S_n$.
Keywords: Cayley graph, Star graph, symmetric group, graph spectrum, eigenvalues, multiplicity. 
@article{SEMR_2016_13_a66,
     author = {S. V. Avgustinovich and E. N. Khomyakova and E. V. Konstantinova},
     title = {Multiplicities of eigenvalues of the {Star} graph},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1258--1270},
     publisher = {mathdoc},
     volume = {13},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2016_13_a66/}
}
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S. V. Avgustinovich; E. N. Khomyakova; E. V. Konstantinova. Multiplicities of eigenvalues of the Star graph. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 13 (2016), pp. 1258-1270. http://geodesic.mathdoc.fr/item/SEMR_2016_13_a66/