Related papers: One more proof about the spectrum of Transposition…
Transposition graph $T_n$ is defined as a Cayley graph over the symmetric group generated by all transpositions. It is known that all eigenvalues of $T_n$ are integers. However, an explicit description of the spectrum is unknown. In this…
Transposition graph $T_n$ is defined as a Cayley graph over the symmetric group generated by all transpositions. It is known that all eigenvalues of $T_n$ are integers. Moreover, zero is its eigenvalue for any $n\geqslant 4$. But the exact…
Recently in graph theory several authors have studied the spectrum of the Cayley graph of the symmetric group S_n generated by the transpositions (1, i) for 2 <= i <= n. Several conjectures were made and partial results were obtained. The…
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…
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…
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…
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 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 $T(S)$…
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},…
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…
We show that the star graph defined as the Cayley graph of ${\mathfrak S}_{n+1}$ generated by the star transpositions is an ${\mathfrak S}_n$-cover of the complete graph $K_{n+1}$, which is known to have fine spectral properties. In the…
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…
This paper deals with the Cayley graph $\mathrm{Cay}(\mathrm{Sym}_n,T_n),$ where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research 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…
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…
Let $S$ be a set of transpositions such that the girth of the transposition graph of $S$ is at least 5. It is shown that the automorphism group of the Cayley graph of the permutation group $H$ generated by $S$ is the semidirect product…
Signal analysis on graphs relies heavily on the graph Fourier transform, which is defined as the projection of a signal onto an eigenbasis of the associated shift operator. Large graphs of similar structure may be represented by a graphon.…
We completely determine the spectrum of an $I$-graph, that is, the eigenvalues of its adjacency matrix. We apply our result to prove known characterizations of connectedness and bipartiteness in $I$-graphs by using an spectral approach.…
A $\mathbb{T}$-gain graph is a simple graph in which a unit complex number is assigned to each orientation of an edge, and its inverse is assigned to the opposite orientation. The associated adjacency matrix is defined canonically, and is…