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 eigenvalues of the Laplacian of $\Cay(\cG)$ have a simple expression in terms of the irreducible characters of transpositions, and of the Littlewood-Richardson coefficients. As a consequence we can prove that the Laplacians of $\cG$ and of $\Cay(\cG)$ have the same first nontrivial eigenvalue. This is equivalent to saying that Aldous's conjecture, asserting that the random walk and the interchange process have the same spectral gap, holds for complete multipartite graphs.
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 $(i,j) \in S$. In the present note, it is proved that two transposition graphs are isomorphic if and only if the corresponding two Cayley graphs are isomorphic. It is also proved that the transposition graph $T(S)$ is edge-transitive if and only if the Cayley graph $Cay(S_n,S)$ is edge-transitive.
Let $d,n$ be positive integers such that $d<n$, and let $X^d_n$ be a Cayley graph generated by a transposition tree of diameter $d$. It is known that every $X^d_n$ with $d<3$ splits into efficient dominating sets. The main result of this paper is that $X^3_n$ does not have efficient dominating sets.
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 $n-1$ (except 0 if $n=2$ or $n=3$).
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 ${\rm{Aut}}(\Cay)$ is the product of the right translation group by $\textsf{N}\rtimes \textsf{D}_{n+1},$ where $\textsf{N}$ is the subgroup fixing $S_n$ element-wise and $\textsf{D}_{n+1}$ is a dihedral group of order $2(n+1)$. We conjecture that $\textsf{N}$ is trivial. We also prove that the subgraph $Γ$ with vertex-set $S_n$ is a $2(n-2)$-regular graph whose aut…
The complete transposition graph is defined to be the graph whose vertices are the elements of the symmetric group $S_n$, and two vertices $α$ and $β$ are adjacent in this graph iff there is some transposition $(i,j)$ such that $α=(i,j) β$. Thus, the complete transposition graph is the Cayley graph $\Cay(S_n,S)$ of the symmetric group generated by the set $S$ of all transpositions. An open problem in the literature is to determine which Cayley graphs are normal. It was shown recently that the Cayley graph generated by 4 cyclically adjacent transpositions is not normal. In the present paper, it…
Let $Γ$ be a Cayley graph generated by a transposition tree. A natural problem is to understand how the properties of the Cayley graph depend on those of the underlying transposition tree. We focus here on diameter and distance related questions. We examine some related bounds and prove some properties about them. This paper has been withdrawn and a new version with these results and further extensions is at abs/1111.3114
The cyclic diagnosability of Cayley graphs generated by transposition triangle-free unicyclic graphs under the PMC and MM* models
School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）；Key Laboratory for Computation and Simulation in Science and Engineering（科学与工程计算模拟重点实验室）；National Center for Applied Mathematics in Hunan（湖南省应用数学中心）
In this paper we study random induced subgraphs of Cayley graphs of the symmetric group induced by an arbitrary minimal generating set of transpositions. A random induced subgraph of this Cayley graph is obtained by selecting permutations with independent probability, $λ_n$. Our main result is that for any minimal generating set of transpositions, for probabilities $λ_n=\frac{1+ε_n}{n-1}$ where $n^{-{1/3}+δ}\le ε_n<1$ and $δ>0$, a random induced subgraph has a.s. a unique largest component of size $\wp(ε_n)\frac{1+ε_n}{n-1}n!$, where $\wp(ε_n)$ is the survival probability of a specific branchi…
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 $T(S)$ whenever $(i,j) \in S$. We prove that if the girth of the transposition graph $T(S)$ is at least 5, then the automorphism group of the Cayley graph $\Cay(S_n,S)$ is the semidirect product $R(S_n) \rtimes \Aut(S_n,S)$, where $\Aut(S_n,S)$ is the set of automorphisms of $S_n$ that fixes $S$. This strengthens a result of Feng on transposition graphs that are tr…
For a graph $G = (V, E)$, a subset $F\subset V(G)$ is called an $R_k$-vertex-cut of $G$ if $G -F$ is disconnected and each vertex $u \in V(G)- F$ has at least $k$ neighbors in $G -F$. The $R_k$-vertex-connectivity of $G$, denoted by $κ^k(G)$, is the cardinality of the minimum $R_k$-vertex-cut of $G$, which is a refined measure for the fault tolerance of network $G$. In this paper, we study $κ^2$ for Cayley graphs generated by $k$-trees. Let $Sym(n)$ be the symmetric group on $\{1, 2, \cdots ,n\}$ and $\mathcal{T}$ be a set of transpositions of $Sym(n)$. Let $G(\mathcal{T})$ be the graph on $n$…
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 $R(H) \rtimes \Aut(H,S)$, where $R(H)$ is the right regular representation of $H$ and $\Aut(H,S)$ is the set of automorphisms of $H$ that fixes $S$ setwise. Furthermore, if the connected components of the transposition graph of $S$ are isomorphic to each other, then $\Aut(H,S)$ is isomorphic to the automorphism group of the line graph of the transposition graph …
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 Bioinformatics. As the main result, we prove that Aut$(\mathrm{Cay}(\mathrm{Sym}_n,T_n))$ is the product of the left translation group by a dihedral group $\mathsf{D}_{n+1}$ of order $2(n+1)$. The proof uses several properties of the subgraph $Γ$ of $\mathrm{Cay}(\mathrm{Sym}_n,T_n)$ induced by the set $T_n$. In particular, $Γ$ is a $2(n-2)$-regular graph whose automo…
Given graphs $G$ and $H$, the generalized Turán number ${\rm ex}(G,H)$ is the maximum number of edges in an $H$-free subgraph of $G$. In this paper, we obtain an asymptotic upper bound on ${\rm ex}(CT_n,C_{2l})$ for any $n \ge 3$ and $l\geq2$, where $C_{2l}$ is the cycle of length $2l$ and $CT_n$ is the complete transposition graph which is defined as the Cayley graph on the symmetric group ${\rm S}_n$ with respect to the set of all transpositions of ${\rm S}_n$.
A problem of practical and theoretical interest is to determine or estimate the diameter of various families of Cayley networks. The previously known estimate for the diameter of Cayley graphs generated by transposition trees is an upper bound given in the oft-cited paper of Akers and Krishnamurthy (1989). In this work, we first assess the performance of their upper bound. We show that for every $n$, there exists a tree on $n$ vertices, such that the difference between the upper bound and the true diameter value is at least $n-4$. Evaluating their upper bound takes time $Ω(n!)$. In this paper,…
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 paper we prove that for any integer $k\geqslant 0$ there exists $n_0$ such that for any $n\geqslant n_0$ and any $m \in \{0, \dots, k\}$, $m$ is an eigenvalue of $T_n$. In particular, it is proved that zero is an eigenvalue of $T_n$ for any $n\neq2$, and one is an eigenvalue of $T_n$ for any odd $n\geqslant 7$ and for any even $n \geqslant 14$. We also present…
The characterization of distance-regular Cayley graphs originated from the problem of identifying strongly regular Cayley graphs, or equivalently, regular partial difference sets. In this paper, a classification of distance-regular Cayley graphs on dicyclic groups is obtained. More specifically, it is shown that every distance-regular Cayley graph on a dicyclic group is a complete graph, a complete multipartite graph, or a non-antipodal bipartite distance-regular graph with diameter $3$ satisfying some additional conditions.
Aldous' spectral gap conjecture states that the second largest eigenvalue of any connected Cayley graph on the symmetric group Sn with respect to a set of transpositions is achieved by the standard representation of Sn. This celebrated conjecture, which was proved in its general form in 2010, has inspired much interest in searching for other families of Cayley graphs on Sn with the property that the largest eigenvalue strictly smaller than the degree is attained by the standard representation of Sn. In this paper, we prove three results on normal Cayley graphs on Sn possessing this property fo…
Let $Γ$ be a Cayley graph of the permutation group generated by a transposition tree $T$ on $n$ vertices. In an oft-cited paper \cite{Akers:Krishnamurthy:1989} (see also \cite{Hahn:Sabidussi:1997}), it is shown that the diameter of the Cayley graph $Γ$ is bounded as $$\diam(Γ) \le \max_{π\in S_n}{c(π)-n+\sum_{i=1}^n \dist_T(i,π(i))},$$ where the maximization is over all permutations $π$, $c(π)$ denotes the number of cycles in $π$, and $\dist_T$ is the distance function in $T$. In this work, we first assess the performance (the sharpness and strictness) of this upper bound. We show that the upp…
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 purpose of this note is to point out that, as mentioned also by P. Renteln, this problem is actually already solved in another context. Indeed it is equivalent to studying the spectrum of so-called Jucys-Murphy elements in the algebra of the symmetric group, which is well understood. The aforementioned conjectures are direct consequences of the existing theor…
The modified bubble-sort graph of dimension $n$ is the Cayley graph of $S_n$ generated by $n$ cyclically adjacent transpositions. In the present paper, it is shown that the automorphism group of the modified bubble sort graph of dimension $n$ is $S_n \times D_{2n}$, for all $n \ge 5$. Thus, a complete structural description of the automorphism group of the modified bubble-sort graph is obtained. A similar direct product decomposition is seen to hold for arbitrary normal Cayley graphs generated by transposition sets.
These lecture notes provide an introduction to automorphism groups of graphs. Some special families of graphs are then discussed, especially the families of Cayley graphs generated by transposition sets.
The unitary Cayley graph of $\mathbb{Z}/n\mathbb{Z}$, denoted $X_n$, is the graph on $\{0,\dots,n-1\}$ where vertices $a$ and $b$ are adjacent if and only if $\gcd(a-b,n) = 1$. We answer a question of Defant and Iyer by constructing a family of infinitely many integers $n$ such that $γ_t(X_n) \leq g(n) - 2$, where $γ_t$ denotes the total domination number and $g$ denotes the Jacobsthal function. We determine the irredundance number, domination number, and lower independence number of certain direct products of complete graphs and give bounds for these parameters for any direct product of compl…
A major problem in the study of combinatorial aspects of permutation groups is to determine the distances in the symmetric group $\Sym_n$ with respect to a generator set. One well-known such a case is when the generator set $S_n$ consists of block transpositions. It should be noted that "the block transposition distance of a permutation" is the distance of the permutation from the identity permutation in the Cayley graph $\Cay$, and "sorting a permutation by block transpositions" is equivalent to finding shortest paths in $\Cay$. The original results in our thesis concern the lower and upper b…
A meander of order n is a simple closed curve in the plane which intersects a horizontal line transversely at 2n points. (Meanders which differ by an isotopy of the line and plane are considered equivalent.) Let Gamma_n be the Cayley graph of the symmetric group S_n as generated by all (n choose 2) transpositions. Let Lambda_n be any interval of maximal length in Gamma_n; this graph is the Hasse diagram of the lattice of noncrossing partitions. The meanders of order n are in one-to-one correspondence with ordered pairs of maximally separated vertices of Lambda_n.
These lecture notes are on automorphism groups of Cayley graphs and their applications to optimal fault-tolerance of some interconnection networks. We first give an introduction to automorphisms of graphs and an introduction to Cayley graphs. We then discuss automorphism groups of Cayley graphs. We prove that the vertex-connectivity of edge-transitive graphs is maximum possible. We investigate the automorphism group and vertex-connectivity of some families of Cayley graphs that have been considered for interconnection networks; we focus on the hypercubes, folded hypercubes, Cayley graphs gener…
Given a finite transitive permutation group $G\leq \operatorname{Sym}(Ω)$, with $|Ω|\geq 2$, the derangement graph $Γ_G$ of $G$ is the Cayley graph $\operatorname{Cay}(G,\operatorname{Der}(G))$, where $\operatorname{Der}(G)$ is the set of all derangements of $G$. Meagher et al. [On triangles in derangement graphs, {\it J. Combin. Theory Ser. A}, 180:105390, 2021] recently proved that $\operatorname{Sym}(2)$ acting on $\{1,2\}$ is the only transitive group whose derangement graph is bipartite and any transitive group of degree at least three has a triangle in its derangement graph. They also sh…
Packing and covering problems for metric spaces, and graphs in particular, are of essential interest in combinatorics and coding theory. They are formulated in terms of metric balls of vertices. We consider a new problem in graph theory which is also based on the consideration of metric balls of vertices, but which is distinct from the traditional packing and covering problems. This problem is motivated by applications in information transmission when redundancy of messages is not sufficient for their exact reconstruction, and applications in computational biology when one wishes to restore an…
The Star graph $S_n$, $n\ge 3$, is the Cayley graph on the symmetric group $Sym_n$ generated by the set of transpositions $\{(12),(13),\ldots,(1n)\}$. In this work we study eigenfunctions of $S_n$ corresponding to the second largest eigenvalue $n-2$. For $n\ge 8$ and $n=3$, we find the minimum cardinality of the support of an eigenfunction of $S_n$ corresponding to the second largest eigenvalue and obtain a characterization of eigenfunctions with the minimum cardinality of the support.
The unitary Cayley graph of $\mathbb Z/n\mathbb Z$, denoted $G_{\mathbb Z/n\mathbb Z}$, is the graph with vertices $0,1,\ldots,$ $n-1$ in which two vertices are adjacent if and only if their difference is relatively prime to $n$. These graphs are central to the study of graph representations modulo integers, which were originally introduced by Erdős and Evans. We give a brief account of some results concerning these beautiful graphs and provide a short proof of a simple formula for the number of cliques of any order $m$ in the unitary Cayley graph $G_{\mathbb Z/n\mathbb Z}$. This formula invol…
We consider the symmetric group $\mathrm{Sym}_n,\,n\geqslant 2$, generated by the set $S$ of transpositions $(1~i),\,2 \leqslant i \leqslant n$, and the Cayley graph $S_n=Cay(\mathrm{Sym}_n,S)$ called the Star graph. For any positive integers $n\geqslant 3$ and $m$ with $n > 2m$, we present a family of $PI$-eigenfunctions of $S_n$ with eigenvalue $n-m-1$. We establish a connection of these functions with the standard basis of a Specht module. In the case of largest non-principal eigenvalue $n-2$ we prove that any eigenfunction of $S_n$ can be reconstructed by its values on the second neighbour…
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 data analysis, as its vertices represent permutations, while the generating set formalizes a notion of distance between rankings. Taking advantage of the rich theory of representations of the symmetric group, we study a particular class of frames, called Frobenius-Schur frames, where every atom belongs to the coefficient space of only one irreducible repres…
The conditional diagnosability and the 2-extra connectivity are two important parameters to measure ability of diagnosing faulty processors and fault-tolerance in a multiprocessor system. The conditional diagnosability $t_c(G)$ of $G$ is the maximum number $t$ for which $G$ is conditionally $t$-diagnosable under the comparison model, while the 2-extra connectivity $κ_2(G)$ of a graph $G$ is the minimum number $k$ for which there is a vertex-cut $F$ with $|F|=k$ such that every component of $G-F$ has at least $3$ vertices. A quite natural problem is what is the relationship between the maximum …
CayleyPy Growth: Efficient growth computations and hundreds of new conjectures on Cayley graphs (Brief version)
Institut Curie, CNRS UMR168, Paris, France（居里研究所，法国国家科学研究中心UMR168，法国巴黎）；University of Washington（华盛顿大学）；Three Gorges Mathematical Research Center, China Three Gorges University（三峡数学研究中心，三峡大学）；Sobolev Institute of Mathematics（索博列夫数学研究所）；Novosibirsk State University（新西伯利亚国立大学）；Independent Researcher（独立研究员）；Moscow State University（莫斯科国立大学）；Kazakh-British Technical University（哈萨克-英国技术大学）；Neapolis University Pafos, Cyprus（尼阿波利斯大学帕福斯校区，塞浦路斯）；City St George's, University of London（伦敦大学城市圣乔治学院）；Hebrew University of Jerusalem（耶路撒冷希伯来大学）；ITMO University（ITMO大学）；Moscow Center for Advanced Studies（莫斯科高等研究中心）；National Research University Higher School of Economics（国立研究大学高等经济学院）；Ariel University（阿里埃勒大学）；Saint-Petersburg Pasteur Institute（圣彼得堡巴斯德研究所）；Criteo AI Lab, Berlin, Germany（Criteo人工智能实验室，德国柏林）；University of Bologna（博洛尼亚大学）；Faculty of Bioengineering and Bioinformatics, Lomonosov Moscow State University（莫斯科国立大学生物工程与生物信息学系）；Researcher at Banner Stat（Banner Stat研究员）；MIREA — Russian Technological University（MIREA — 俄罗斯技术大学）；Research Center of the Artificial Intelligence Institute, Innopolis University（因诺波利斯大学人工智能研究所研究中心）；Inpdependent Researcher（独立研究员）；LIX, Ecole Polytechnique/CNRS, IP Paris（LIX，巴黎综合理工学院/法国国家科学研究中心，巴黎理工学院）
A conjecture by D. Aldous, which can be formulated as a statement about the first nontrivial eigenvalue of the Laplacian of certain Cayley graphs on the symmetric group generated by transpositions, has been recently proven by Caputo, Liggett and Richthammer. Their proof is a subtle combination of two ingredients: a nonlinear mapping in the group algebra of the symmetric groups which permits a proof by induction, and a quite hard estimate named the octopus inequality. In this paper we present a simpler and more transparent proof of the octopus inequality, which emerges naturally when looking at…
We prove an isoperimetric inequality for conjugation-invariant sets of size $k$ in $S_n$, showing that these necessarily have edge-boundary considerably larger than some other sets of size $k$ (provided $k$ is small). Specifically, let $T_n$ denote the Cayley graph on $S_n$ generated by the set of all transpositions. We show that if $A \subset S_n$ is a conjugation-invariant set with $|A| = pn! \leq n!/2$, then the edge-boundary of $A$ in $T_n$ has size at least $$c \cdot \frac {\log_2 (\tfrac 1{p})}{\log_2 \log_2 (\tfrac 2{p})}\cdot n \cdot |A|,$$ where $c$ is an absolute constant. (This is s…
A vertex triple $(u,v,w)$ of a graph is called a $2$-geodesic if $v$ is adjacent to both $u$ and $w$ and $u$ is not adjacent to $w$. A graph is said to be $2$-geodesic transitive if its automorphism group is transitive on the set of $2$-geodesics. In this paper, a complete classification of $2$-geodesic transitive graphs of order $p^n$ is given for each prime $p$ and $n\leq 3$. It turns out that all such graphs consist of three small graphs: the complete bipartite graph $K_{4,4}$ of order $8$, the Schläfli graph of order $27$ and its complement, and fourteen infinite families: the cycles $C_p,…
Parametric families in the centre ${\bf Z}({\bf C}[S_n])$ of the group algebra of the symmetric group are obtained by identifying the indeterminates in the generating function for Macdonald polynomials as commuting Jucys-Murphy elements. Their eigenvalues provide coefficients in the double Schur function expansion of 2D Toda $τ$-functions of hypergeometric type. Expressing these in the basis of products of power sum symmetric functions, the coefficients may be interpreted geometrically as parametric families of quantum Hurwitz numbers, enumerating weighted branched coverings of the Riemann sph…
For a permutation group $G$ acting on a set $V$, a subset $I$ of $G$ is said to be an intersecting set if for every pair of elements $g,h\in I$ there exists $v \in V$ such that $g(v) = h(v)$. The intersection density $ρ(G)$ of a transitive permutation group $G$ is the maximum value of the quotient $|I|/|G_v|$ where $G_v$ is a stabilizer of a point $v\in V$ and $I$ runs over all intersecting sets in $G$. If $G_v$ is the largest intersecting set in $G$ then $G$ is said to have the Erdős-Ko-Rado (EKR)-property, and moreover, $G$ has the strict-EKR-property if every intersecting set of maximum siz…
The hypercontractive inequality is a fundamental result in analysis, with many applications throughout discrete mathematics, theoretical computer science, combinatorics and more. So far, variants of this inequality have been proved mainly for product spaces, which raises the question of whether analogous results hold over non-product domains. We consider the symmetric group, $S_n$, one of the most basic non-product domains, and establish hypercontractive inequalities on it. Our inequalities are most effective for the class of \emph{global functions} on $S_n$, which are functions whose $2$-norm…
Let $S_n$ denote the symmetric group on $n$ elements, and $Σ\subseteq S_{n}$ a symmetric subset of permutations. Aldous' spectral gap conjecture, proved by Caputo, Liggett and Richthammer [arXiv:0906.1238], states that if $Σ$ is a set of transpositions, then the second eigenvalue of the Cayley graph $\mathrm{Cay}\left(S_{n},Σ\right)$ is identical to the second eigenvalue of the Schreier graph on $n$ vertices depicting the action of $S_{n}$ on $\left\{ 1,\ldots,n\right\}$. Inspired by this seminal result, we study similar questions for other types of sets in $S_{n}$. Specifically, we consider n…
This is an overview of recent results on the use of 2D Toda $τ$-functions as generating functions for multiparametric families of weighted Hurwitz numbers. The Bose-Fermi equivalence composed with the characteristic map provides an isomorphism between the zero charge sector of the Fermionic Fock space and the direct sum of the centers of the group algebra of the symmetric groups $S_n$. Specializing the fermionic formula to the case of diagonal group elements gives $τ$-functions of hypergeometric type, for which the expansion over products of Schur functions is diagonal, with coefficients of {\…
