On the partitions associated with the smallest eigenvalues of certain Cayley graphs on symmetric group generated by cycles
Introduction
Let G be a finite group and S be an inverse closed subset of G, i.e., and . The Cayley graph is the graph which has the elements of G as its vertices and two vertices are joined by an edge if and only if for some .
A Cayley graph is said to be normal if S is closed under conjugation. It is well known that the eigenvalues of a normal Cayley graph 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 are given by where χ ranges over all the irreducible characters of G. Moreover, the multiplicity of is .
Let be the symmetric group on and 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 are integers.
Corollary 1.2 The eigenvalues of a normal Cayley graph are integers.
We note here that if S is not closed under conjugation, then the eigenvalues of may not be integers [7].
Let and be the set of all s cycles in which do not fix 1, i.e. Abdollahi and Vatandoost [1] conjectured that the eigenvalues of are integers, and contains all integers in the range from to (with the sole exception that when or 3, zero is not an eigenvalue of ). Krakovski and Mohar [13] proved that for and each integer , are eigenvalues of with multiplicity at least . Furthermore, if , then 0 is an eigenvalue of with multiplicity at least . Later, Chen, Ghorbani, and Wong [4] showed that the eigenvalues of 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 and where A better understanding of the Cayley graph is important to the study of the Cayley graph .
The transposition network of order n! is the Cayley graph . Note that 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 was obtained as well. Now, we look at the Cayley graph . Since 3-cycle is an even permutation, is disconnected with two isomorphic connected components of all even permutations and all odd permutations. In fact, is the disjoint union of two copies of , where is the Alternating group of degree n. So, it is quite natural to study the Cayley graph . Partial ordering of was studied by Mühle and Nadeau [18]. However, not much is known about the eigenvalues of . Nevertheless, we have the following obvious lemma.
Lemma 1.3 e is an eigenvalue of with multiplicity m if and only if e is an eigenvalue of with multiplicity 2m.
A partition of n is a weakly decreasing sequence such that . We write . Each is called the i-th part of the partition and the size of λ is defined to be n and denoted by . For convenience, the notation is used to denote the partition where each is the distinct nonzero parts that occur with multiplicity . For example, Given a partition , 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 boxes for . The boxes also are called cells. The box in row i and column j is labelled by coordinate and we call this box the cell . 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 and it is called the transpose of λ.
It is well known that both the conjugacy classes of and the irreducible characters of are indexed by partitions λ of . For each , the Specht module is an irreducible -module (see [20, Section 2.3]). Let be the character corresponding to the Specht module . Then, the dimension of is .
By Theorem 1.1, the eigenvalues of are given by where is a s-cycle in and λ ranges over all the partitions of n. Moreover, the multiplicity of is . Throughout this paper, the symbol is referred to the eigenvalue of labelled by the partition λ of .
Let be the smallest eigenvalue of . Then, Let A partition λ is said to be associated to the smallest eigenvalue of if .
Problem 1.4 Find all the partitions associated to the smallest eigenvalue of .
Note that the smallest eigenvalue of a connected bipartite d-regular graph is equal to −d with multiplicity 1. Since is a bipartite -regular graph, is the smallest eigenvalue of 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 . Therefore, is the only possibility for partition of n which is associated to the smallest eigenvalue . Therefore, Problem 1.4 is settled for .
In this paper, we will settle the case in Problem 1.4. Specifically, we will prove the following theorem.
Theorem 1.5 Let ν and b be the unique integers such that and . A partition λ of n is associated to the smallest eigenvalue of if and only if or where
Section snippets
Smallest eigenvalues of
Let . By [8, Equation (5.2)], for all . Therefore, it follows from Equation (1) that We have proved the following theorem.
Theorem 2.1 The eigenvalues of the Cayley graph are given by where (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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