Catalogue of the Star graph eigenvalue multiplicities

The Star graph Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_n$$\end{document}, n⩾2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\geqslant 2$$\end{document}, is the Cayley graph over the symmetric group Symn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {Sym}_n$$\end{document} generated by transpositions (1i),2⩽i⩽n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1~i),\,2\leqslant i \leqslant n$$\end{document}. This set of transpositions plays an important role in the representation theory of the symmetric group. The spectrum of Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_n$$\end{document} contains all integers from -(n-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-(n-1)$$\end{document} to n-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n-1$$\end{document}, and also zero for n⩾4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\geqslant 4$$\end{document}. In this paper we observe methods for getting explicit formulas of eigenvalue multiplicities in the Star graphs Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_n$$\end{document}, present such formulas for the eigenvalues ±(n-k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pm (n-k)$$\end{document}, where 2⩽k⩽12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\leqslant k \leqslant 12$$\end{document}, and finally collect computational results of all eigenvalue multiplicities for n⩽50\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\leqslant 50$$\end{document} in the catalogue.


Introduction
The Star graph S n , n 2, is the Cayley graph over the symmetric group Sym n of permutations π = [π 1 π 2 . . . π n ] with the generating set {(1 i) ∈ Sym n : 2 i n} of all transpositions (1 i) swapping the 1st and the ith elements of a permutation π. It is a connected bipartite (n − 1)-regular graph of order n! and diameter diam(S n ) = 3(n−1) 2 [3]. A graph is integral if all eigenvalues of its adjacency matrix are integers. In 1974, Harary and Schwenk [10] posed a question on graphs having integral spectra. In general, most of the graphs have nonintegral eigenvalues [2].
In 2000, Friedman [8] investigated the second smallest non-negative eigenvalue λ 2 of Cayley graphs on the symmetric group generated by transpositions. He proved that among all sets of n − 1 transpositions which generate the symmetric group, the set whose associated Cayley graph has λ 2 = 1 is the set This means that there are no other integral Cayley graphs over the symmetric group generated by sets of n − 1 transpositions.
In 2009, Abdollahi and Vatandoost conjectured [1] that the spectrum of S n is integral, and contains all integers in the range from −(n − 1) up to n − 1 (with the sole exception that when n 3, zero is not an eigenvalue of S n ). Partially this conjecture was based on the known fact about the spectrum of an r -regular graph which lies in the segment [−r, r ] [5]. They verified this conjecture numerically using GAP for n 6. In 2012, Krakovski and Mohar [14] proved the second part of the conjecture. More precisely, they proved that for n 2 and for each integer 1 k n − 1, the values ±(n − k) are eigenvalues of S n with multiplicity at least n−2 k−1 . If n 4, then 0 is an eigenvalue of S n with multiplicity at least n−1 2 . Since the Star graph is bipartite, mul(n − k) = mul(−n + k) for each integer 1 k n. Moreover, ±(n − 1) are simple eigenvalues of S n .
At the same time, Chapuy and Feray [6] showed that the integrality of the Star graphs was already solved in another context, since it is equivalent to studying the spectrum of Jucys-Murphy elements in the algebra of the symmetric group [11]. This connection between two kinds of spectra implies that the Star graph is integral. References on the topic can be also found in the introduction of the paper by Renteln [15].
A lower bound on multiplicities of eigenvalues of S n given by Krakovski and Mohar was improved by Chapuy and Feray as follows: In 2016, Avgustinovich et al. [4,12] suggested a method for getting explicit formulas for multiplicities of eigenvalues ±(n − k) in the Star graphs S n and presented such formulas for 2 k 5. Moreover, a lower bound on multiplicity of eigenvalues of S n for sufficiently large n was obtained. It was proved that for a fixed integer eigenvalue of the Star graph S n , its multiplicity is at least 2 1 2 n log n(1−o(1)) [4]. In 2018, Khomyakova [13] investigated the behavior of the eigenvalues multiplicity function of the Star graph S n for eigenvalues ±(n−k) where 1 k n+1 2 . It was shown that the function has a polynomial behavior in n. Moreover, explicit formulas for calculating multiplicities of eigenvalues ±(n − k) where 2 k 12 were also presented in the paper. Computational results showed that the same polynomial behavior of the eigenvalues multiplicity function occurs for any integers n 2 and 1 k n.
In this paper, we review methods used for getting explicit formulas for eigenvalue multiplicities in the Star graphs S n , present these formulas for the eigenvalues ±(n − k), where 2 k 12, and finally collect computational results of all eigenvalue multiplicities for n 50 in the catalogue provided in the electronic supplementary material.

Theoretical results
To describe a combinatorial approach for calculating multiplicities of eigenvalues of the Star graphs S n , n 2, we need to give basic definitions and notation on representation theory of the symmetric group [16].
The symmetric group Sym n consists of all bijections of {1, 2, . . . , n} to itself using compositions as the multiplication. For any permutation π ∈ Sym n , we view its cycle type as a partition.
Let λ is a partition of n. A Young tableau of shape λ is obtained by filling in the boxes of a Young diagram of λ with the elements {1, 2, . . . , n}, where each number occurring exactly once. Thus, the Young tableau of shape λ is the set where m ∈ {1, . . . , n} and i, j are the ordinate and the abscissa of the box containing m, correspondingly. A standard Young tableau is a Young tableau whose the entries are increasing across each row and each column.
We write λ for the conjugate partition of λ defined by λ = (λ 1 , λ 2 , . . . , λ l ), where l = λ 1 , λ j = max{ j : . Then, the hook length h i j is defined by the following formula: Now let us show relationships between standard Young tableaux and eigenvalue multiplicities of the Star graphs.
Let G be a group and V be a finite-dimensional vector space over the complex numbers. Let G L(V ) stands for the set of all invertible linear transformations of V to itself, called the general linear group of V . Then a representation of G on V is a group homomorphism ρ : G → G L(V ), and V is a vector space of the representation with dimension dim(V ). The representation is irreducible if it has no proper subspace closed under the action of ρ. Two representations ρ 1 : G → G L(V 1 ) and ρ 2 : G → G L(V 2 ) are equivalent if there exists a bijective linear map ϕ : V 1 → V 2 such that ϕρ 2 (g) = ρ 1 (g)ϕ for all g ∈ G.
The symmetric group Sym n has order n!, its conjugacy classes are labeled by partitions of n, and according to the representation theory of a finite group, the set of inequivalent irreducible representations is defined by partitions of n. We denote by V λ a vector space of the irreducible representation associated with the partition λ n. It is known [16] that and the following equality holds [6] mul where I λ (n − k) is the number of standard Young tableaux of shape λ satisfying c(n) = n − k. Since dim(V λ ) is equal to the number of all partitions of shape λ, i.e., the number of standard Young tableaux, it is calculated by the Hook Formula [7]: where λ n. Let A k be the set of partitions of n of length l = n − k + 1 with the last element 1. For any λ = (λ 1 , . . . , λ n−k , 1), letλ be a partition (λ 1 , . . . , λ n−k ) of n − 1. Then the following result holds.
Theorem 2.2 [13] Let n, k ∈ Z, n 2 and 1 k n+1 2 , then the multiplicity mul(n − k) of eigenvalue (n − k) of the Star graph S n is calculated by the formula: where P(n) is a polynomial of degree 2k − 3.
Computational results show that theorem holds for any n 2 and 1 k n.