Elsevier

Linear Algebra and its Applications

Volume 630, 1 December 2021, Pages 179-203
Linear Algebra and its Applications

On the partitions associated with the smallest eigenvalues of certain Cayley graphs on symmetric group generated by cycles

https://doi.org/10.1016/j.laa.2021.08.001Get rights and content

Abstract

Let Sn be the symmetric group on [n]={1,2,,n} and Zn(s)={αSn:αis ans-cycle} where 2sn. In this paper, we determine all the partitions associated with the smallest eigenvalues of the Cayley graph Γ(Sn,Zn(s)) for s=3.

Introduction

Let G be a finite group and S be an inverse closed subset of G, i.e., 1S and aSa1S. The Cayley graph Γ(G,S) is the graph which has the elements of G as its vertices and two vertices u,vG are joined by an edge if and only if v=au for some aS.

A Cayley graph Γ(G,S) is said to be normal if S is closed under conjugation. It is well known that the eigenvalues of a normal Cayley graph Γ(G,S) can be expressed in terms of the irreducible characters of G [5, p. 235 ].

Theorem 1.1 [2], [6], [17], [19]

The eigenvalues of a normal Cayley graph Γ(G,S) are given byηχ=1χ(1)aSχ(a), where χ ranges over all the irreducible characters of G. Moreover, the multiplicity of ηχ is χ(1)2.

Let Sn be the symmetric group on [n]={1,,n} and SSn be closed under conjugation. Since central characters are algebraic integers ([9, Theorem 3.7 on p. 36]) and that the characters of the symmetric group are integers ([9, 2.12 on p. 31] or [21, Corollary 2 on p. 103]), by Theorem 1.1, the eigenvalues of Γ(Sn,S) are integers.

Corollary 1.2

The eigenvalues of a normal Cayley graph Γ(Sn,S) are integers.

We note here that if S is not closed under conjugation, then the eigenvalues of Γ(Sn,S) may not be integers [7].

Let 2sn and Cn(s) be the set of all s cycles in Sn which do not fix 1, i.e.Cn(s)={αSn:α(1)1and α is an s-cycle}. Abdollahi and Vatandoost [1] conjectured that the eigenvalues of Γ(Sn,Cn(2)) are integers, and contains all integers in the range from (n1) to n1 (with the sole exception that when n=2 or 3, zero is not an eigenvalue of Γ(Sn,Cn(2))). Krakovski and Mohar [13] proved that for n2 and each integer 1ln1, ±(nl) are eigenvalues of Γ(Sn,Cn(2)) with multiplicity at least (n2l1). Furthermore, if n4, then 0 is an eigenvalue of Γ(Sn,Cn(2)) with multiplicity at least (n12). Later, Chen, Ghorbani, and Wong [4] showed that the eigenvalues of Γ(Sn,Cn(2)) are integers. Chapuy and Féray [3] pointed out that the conjecture could be proved by using Jucys-Murphy elements (see also [11]). Recently, Terry and Wong [15] gave a relationship between the eigenvalues of Γ(Sn,Cn(s)) and Γ(Sn,Zn(s)) whereZn(s)={αSn:α is an s-cycle}. A better understanding of the Cayley graph Γ(Sn,Zn(s)) is important to the study of the Cayley graph Γ(Sn,Cn(s)).

The transposition network of order n! is the Cayley graph Γ(Sn,Zn(2)). Note that Γ(Sn,Zn(2)) is a bipartite graph with vertex sets A and B where A consists of all even permutations and B consists of all odd permutations. The transposition network was first introduced by Lakshmivarahan, Jwo and Dhall [14]. Finding the bisection width of the transposition network is an open question posed by Leighton [16, p. 776]. The question was resolved by Kalpakis and Yesha [12]. Furthermore, the expression for all the eigenvalues of Γ(Sn,Zn(2)) was obtained as well. Now, we look at the Cayley graph Γ(Sn,Zn(3)). Since 3-cycle is an even permutation, Γ(Sn,Zn(3)) is disconnected with two isomorphic connected components of all even permutations and all odd permutations. In fact, Γ(Sn,Zn(3)) is the disjoint union of two copies of Γ(An,Zn(3)), where An is the Alternating group of degree n. So, it is quite natural to study the Cayley graph Γ(An,Zn(3)). Partial ordering of Γ(An,Zn(3)) was studied by Mühle and Nadeau [18]. However, not much is known about the eigenvalues of Γ(An,Zn(3)). Nevertheless, we have the following obvious lemma.

Lemma 1.3

e is an eigenvalue of Γ(An,Zn(3)) with multiplicity m if and only if e is an eigenvalue of Γ(Sn,Zn(3)) with multiplicity 2m.

A partition λ=(λ1,,λr) of n is a weakly decreasing sequence λ1λr1 such that λ1++λr=n. We write λn. Each λi is called the i-th part of the partition and the size of λ is defined to be n and denoted by |λ|. For convenience, the notation (γ1a1,,γrar)n is used to denote the partition where each γi is the distinct nonzero parts that occur with multiplicity ai. For example,(6,6,6,5,5,4,3,3,3,2,1)(63,52,4,33,2,1). Given a partition λ=(λ1,λ2,,λr)n, it can be associated to a Ferrers diagram (see [20, Definition 2.1.1]). Basically, the Ferrers diagram of λ is an array of n boxes having r left-justified row with row i containing λi boxes for 1ir. The boxes also are called cells. The box in row i and column j is labelled by coordinate (i,j) and we call this box the cell (i,j). The Ferrers diagram of λ will just be called λ-diagram. The transpose of a λ-diagram (see [20, Definition 3.5.2]) is also a Ferrers diagram of a partition. The partition is denoted by λT and it is called the transpose of λ.

It is well known that both the conjugacy classes of Sn and the irreducible characters of Sn are indexed by partitions λ of [n]. For each λn, the Specht module Sλ is an irreducible CSn-module (see [20, Section 2.3]). Let χλ be the character corresponding to the Specht module Sλ. Then, the dimension of Sλ is fλ=χλ(1).

By Theorem 1.1, the eigenvalues of Γ(Sn,Zn(s)) are given byηλ(s)=1fλaZn(s)χλ(a)=χλ(α)fλ|Zn(s)|=χλ(α)fλ(n!(ns)!s) where α=(123s) is a s-cycle in Sm and λ ranges over all the partitions of n. Moreover, the multiplicity of ηλ(s) is (fλ)2. Throughout this paper, the symbol ηλ(s) is referred to the eigenvalue of Γ(Sn,Zn(s)) labelled by the partition λ of n=|λ|.

Let M(n,s) be the smallest eigenvalue of Γ(Sn,Zn(s)). Then,M(n,s)=minλnηλ(s). LetP(n,s)={λn:ηλ(s)=M(n,s)}. A partition λ is said to be associated to the smallest eigenvalue of Γ(Sn,Zn(s)) if λP(n,s).

Problem 1.4

Find all the partitions associated to the smallest eigenvalue of Γ(Sn,Zn(s)).

Note that the smallest eigenvalue of a connected bipartite d-regular graph is equal to −d with multiplicity 1. Since Γ(Sn,Zn(2)) is a bipartite n(n1)/2-regular graph, n(n1)/2 is the smallest eigenvalue of Γ(Sn,Zn(2)) with multiplicity 1. There are exactly two partitions of n which correspond to the Specht module of dimension 1. One of them is (n), which corresponds to the eigenvalue n(n1)/2. Therefore, (1n) is the only possibility for partition of n which is associated to the smallest eigenvalue n(n1)/2. Therefore, Problem 1.4 is settled for s=2.

In this paper, we will settle the case s=3 in Problem 1.4. Specifically, we will prove the following theorem.

Theorem 1.5

Let ν and b be the unique integers such that n=ν2+b and 1b2ν+1. A partition λ of n is associated to the smallest eigenvalue of Γ(Sn,Zn(3)) if and only if λ=λ0 or λT=λ0 whereλ0={((ν+1)b,ννb)if1bν;((ν+1)ν,(bν))ifν+1b2ν+1.

Section snippets

Smallest eigenvalues of Γ(Sn,Zn(3))

Let λ=(λ1,λ2,,λr)n. By [8, Equation (5.2)],χλ(β)fλ=1n(n1)(n2)(12i=1r((λii)(λii+1)(2(λii)+1)+i(i1)(2i1))32n(n1)), for all βZn(3). Therefore, it follows from Equation (1) thatηλ(3)=χλ(β)fλ(n!(n3)!3)=13(12i=1r((λii)(λii+1)(2(λii)+1)+i(i1)(2i1))32n(n1)). We have proved the following theorem.

Theorem 2.1

The eigenvalues of the Cayley graph Γ(Sn,Zn(3)) are given byηλ(3)=16i=1r((λii)(λii+1)(2(λii)+1)+i(i1)(2i1))12n(n1), where λ=(λ1,,λr)n (here λ ranges over all the partitions of n).

Declaration of Competing Interest

No competing interest.

Acknowledgement

We would like to thank the anonymous referee for his/her comments that has helped us improve the presentation of this paper.

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