Related papers: A note on a Cayley graph of S_n

A Transposition graph $T_n$ is defined as a Cayley graph over the symmetric group $Sym_n$ generated by all transpositions. This paper shows how the spectrum of $T_n$ can be obtained using the spectral properties of the Jucys-Murphy…

For an integer $n\geq 2$, let $X_n$ be the Cayley graph on the symmetric group $S_n$ generated by the set of transpositions ${(1 2),(1 3),...,(1 n)}$. It is shown that the spectrum of $X_n$ contains all integers from $-(n-1)$ to $n-1$…

Let $S$ be a set of transpositions generating the symmetric group $S_n$. The transposition graph of $S$ is defined to be the graph with vertex set $\{1,\ldots,n\}$, and with vertices $i$ and $j$ being adjacent in $T(S)$ whenever $(i,j) \in…

This paper deals with the Cayley graph $\Cay,$ where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research in Bioinformatics. We prove that…

We investigate subsets of the symmetric group with structure similar to that of a graph. The trees of these subsets correspond to minimal conjugate generating sets of the symmetric group. There are two main theorems in this paper. The first…

The complete transposition graph is defined to be the graph whose vertices are the elements of the symmetric group $S_n$, and two vertices $\alpha$ and $\beta$ are adjacent in this graph iff there is some transposition $(i,j)$ such that…

In this paper we study the Cayley graph $\mathrm{Cay}(S_n,T)$ of the symmetric group $S_n$ generated by a set of transpositions $T$. We show that for $n\geq 5$ the Cayley graph is normal. As a corollary, we show that its automorphism group…

The transposition graph $Cay(S_n,T_n)$ is the Cayley graph on the symmetric group $S_n$ generated by the set $T_n$ of all transpositions. In this paper, we show that each integer in the interval $\left[-{\lfloor(2n+1)/3 \rfloor\choose 2},…

In a recent paper Gunnells, Scott and Walden have determined the complete spectrum of the Schreier graph on the symmetric group corresponding to the Young subgroup $S_{n-2}\times S_2$ and generated by initial reversals. In particular they…

A Transposition graph $T_n$ is defined as a Cayley graph over the symmetric group $Sym_n$ generated by all transpositions. It is known that the spectrum of $T_n$ consists of integers, but it is not known exactly how these numbers are…

An important problem in the field of graph signal processing is developing appropriate overcomplete dictionaries for signals defined on different families of graphs. The Cayley graph of the symmetric group has natural applications in ranked…

Let $S$ be a set of transpositions that generates the symmetric group $S_n$, where $n \ge 3$. The transposition graph $T(S)$ is defined to be the graph with vertex set $\{1,\ldots,n\}$ and with vertices $i$ and $j$ being adjacent in $T(S)$…

Let $S_n$ denote the symmetric group on $n$ elements, and $\Sigma\subseteq S_{n}$ a symmetric subset of permutations. Aldous' spectral gap conjecture, proved by Caputo, Liggett and Richthammer [arXiv:0906.1238], states that if $\Sigma$ is a…

Current methods of graph signal processing rely heavily on the specific structure of the underlying network: the shift operator and the graph Fourier transform are both derived directly from a specific graph. In many cases, the network is…

Given a finite simple graph $\cG$ with $n$ vertices, we can construct the Cayley graph on the symmetric group $S_n$ generated by the edges of $\cG$, interpreted as transpositions. We show that, if $\cG$ is complete multipartite, the…

We study subsets $T$ consisting of some transpositions $(i,j)$ of the symmetric group $S_n$ on $\{1,\dots,n\}$ such that the Cayley graph $\Gamma_T:=Cay(S_n,T)$ is an integral graph, i.e., all eigenvalues of an adjacency matrix of…

Arrangement graphs were introduced for their connection to computational networks and have since generated considerable interest in the literature. In a pair of recent articles by Chen, Ghorbani and Wong, the eigenvalues for the adjacency…

We construct explicit generating sets S_n and \tilde S_n of the for the alternating and the symmetric groups, which turn the Cayley graphs C(Alt(n), S_n) and C(Sym(n), \tilde S_n) into a family of bounded degree expanders for all n. This…

A spline is an assignment of polynomials to the vertices of a graph whose edges are labeled by ideals, where the difference of two polynomials labeling adjacent vertices must belong to the corresponding ideal. The set of splines forms a…

The Laplacian of a (weighted) Cayley graph on the Weyl group $W(B_n)$ is a $N\times N$ matrix with $N = 2^n n!$ equal to the order of the group. We show that for a class of (weighted) generating sets, its spectral gap (lowest nontrivial…