We consider the Cayley graph on the symmetric group Sn generated by derangements. It is well known that the eigenvalues of this grpah are indexed by partitions of n. We investigate how these eigenvalues are determined by the shape of their corresponding partitions. In particular, we show that the sign of an eigenvalue is the parity of the number of cells below the first row of the corresponding Ferrers diagram. We also provide some lower and upper bounds for the absolute values of these eigenvalues.
We give a new recurrence formula for the eigenvalues of the derangement graph. Consequently, we provide a simpler proof of the Alternating Sign Property of the derangement graph. Moreover, we prove that the absolute value of the eigenvalue decreases whenever the corresponding partition decreases in the dominance order. In particular, this settles affirmatively a conjecture of Ku and Wales (J. of Combin. Theory, Series A 117 (2010) 289--312) regarding the lower and upper bound for the absolute values of these eigenvalues.
In this paper, we consider the distance spectra of the derangement graphs. First we give a constructive proof that the connected derangement graphs are of diameter 2. Then we obtain their distance spectra. In particular, we determine all their extremal distance eigenvalues.
We consider the derangement graph in which the vertices are permutations of $\{ 1,\ldots, n\}$. Two vertices are joined by an edge if the corresponding permutations differ in every position. The derangement graph is known to be Hamiltonian and Hamilton-connected. In this note, we show that the derangement graph is edge pancyclic if $n\ge 4$.
A permutation sigma in Sn is a k-derangement if for any subset X = {a1, . . ., ak} \subseteq [n], {sigma(a1), . . ., sigma(ak)} is not equal to X. One can form the k-derangement graph on the set of permutations of Sn by connecting two permutations sigma and tau if sigma(tau)^-1 is a k-derangement. We characterize when such a graph is connected or Eulerian. For n an odd prime power, we determine the independence, clique and chromatic number of the 2-derangement graph.
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…
A derangement of a graph $G=(V,E)$ is an injective function $f:V\to V$ such that for all $v\in V$, $f(v)\neq v$ and $(v,f(v))\in E$. Not all graphs admit a derangement and previous results have characterized graphs with derangements using neighborhood conditions for subsets of $V$. We establish an alternative criterion for the existence of derangements on a graph. We analyze strict Nash equilibria of the biologically motivated Territorial Raider Game, a multi-player competition for resources in a spatially structured population based on animal raiding and defending behavior. We find that a gra…
Let G=PGL(2,q) be the projective general linear group acting on the projective line P_q. A subset S of G is intersecting if for any pair of permutations π,σin S, there is a projective point p in P_q such that p^π=p^σ. We prove that if S is intersecting, then the size of S is no more than q(q-1). Also, we prove that the only sets S that meet this bound are the cosets of the stabilizer of a point of P_q.
We introduce the notion of a graph derangement, which naturally interpolates between perfect matchings and Hamiltonian cycles. We give a necessary and sufficient condition for the existence of graph derangements on a locally finite graph. This result was first proved by W.T. Tutte in 1953 by applying some deeper results on digraphs. We give a new, simple proof which amounts to a reduction to the (Menger-Egervary-Konig-)Hall(-Hall) Theorem on transversals of set systems. Finally, we consider the problem of classifying all cycle types of graph derangements on m x n checkerboard graphs. Our prese…
We show that each perfect matching in a bipartite graph $G$ intersects at least half of the perfect matchings in $G$. This result has equivalent formulations in terms of the permanent of the adjacency matrix of a graph, and in terms of derangements and permutations on graphs. We give several related results and open questions.
Let $f: \{1, ..., n\} \rightarrow \{1, ..., n\}$ be a function (not necessarily one-to-one). An $f-derangement$ is a permutation $ g:\{1,...,n\} \rightarrow \{1,...,n\}$ such that $g(i) \neq f(i)$ for each $ i = 1, ..., n$. When $f$ is itself a permutation, this is a standard derangement. We examine properties of f-derangements, and show that when we fix the maximum number of preimages for any item under $f$, the fraction of permutations that are f-derangements tends to $ 1/e$ for large $n$, regardless of the choice of $f$. We then use this result to analyze a heuristic method to decompose bip…
In previous work, we associated to any finite simple graph a particular set of derangements of its vertices. These derangements are in bijection with the spheres in the wedge sum describing the homotopy type of the boolean complex for this graph. Here we study the frequency with which a given derangement appears in this set.
We study the family of \emph{derangement action digraphs}, which are a subfamily of the group action graphs introduced in [Fred Annexstein, Marc Baumslag, and Arnold L. Rosenberg, Group action graphs and parallel architectures, \emph{SIAM J. Comput.} 19 (1990), no. 3, 544--569]. For any non-empty set $X$ and a non-empty subset $S$ of $\Der(X)$, the set of derangments of $X$, we define the derangement action digraph $\rm\overrightarrow{DA}(X;S)$ to have vertex set $X$, and an arc from $x$ to $y$ if and only if $y=x^s$ for some $s\in S$. In common with Cayley graphs and digraphs, derangement act…
Given a permutation group $G$, the derangement graph $Γ_G$ of $G$ is the Cayley graph with connection set the set of all derangements of $G$. We prove that, when $G$ is transitive of degree at least $3$, $Γ_G$ contains a triangle. The motivation for this work is the question of how large can be the ratio of the independence number of $Γ_G$ to the size of the stabilizer of a point in $G$. We give examples of transitive groups where this ratio is maximum.
In this paper we consider the derangement graph for the group $\textrm{PSU}(3,q)$ where $q$ is a prime power. We calculate all eigenvalues for this derangement graph and use these eigenvalues to prove that $\textrm{PSU}(3,q)$ has the Erdős-Ko-Rado property and, provided that $q\neq 2, 5$, another property that we call the {\textsl Erdős-Ko-Rado module property}.
Let $A_n$ be the alternating group of even permutations of $X:=\{1,2,...,n\}$ and ${\mathcal E}_n$ the set of even derangements on $X.$ Denote by $A\T_n^q$ the tensor product of $q$ copies of $A\T_n,$ where the Cayley graph $A\T_n:=\T(A_n,{\mathcal E}_n)$ is called the even derangement graph. In this paper, we intensively investigate the properties of $A\T_n^q$ including connectedness, diameter, independence number, clique number, chromatic number and the maximum-size independent sets of $A\T_n^q.$ By using the result on the maximum-size independent sets $A\T_n^q$, we completely determine the …
In this paper we prove an Erdős-Ko-Rado-type theorem for intersecting sets of permutations. We show that an intersecting set of maximal size in the projective general linear group PGL(3,q), in its natural action on the points of the projective line, is either a coset of the stabilizer of a point or a coset of the stabilizer of a line.
We construct and analyze an explicit basis for the homology of the boolean complex of a Coxeter system. This gives combinatorial meaning to the spheres in the wedge sum describing the homotopy type of the complex. We assign a set of derangements to any finite simple graph. For each derangement, we construct a corresponding element in the homology of the complex, and the collection of these elements forms a basis for the homology of the boolean complex. In this manner, the spheres in the wedge sum describing the homotopy type of the complex can be represented by a set of derangements. We give a…
Let $G$ be a transitive permutation group of degree $n$. We say that $G$ is $2'$-elusive if $n$ is divisible by an odd prime, but $G$ does not contain a derangement of odd prime order. In this paper we study the structure of quasiprimitive and biquasiprimitive $2'$-elusive permutation groups, extending earlier work of Giudici and Xu on elusive groups. As an application, we use our results to investigate automorphisms of finite arc-transitive graphs of prime valency.
In this paper we give a proof that the largest set of perfect matchings, in which any two contain a common edge, is the set of all perfect matchings that contain a fixed edge. This is a version of the famous Erdős-Ko-Rado theorem for perfect matchings. The proof given in this paper is algebraic, we first determine the least eigenvalue of the perfect matching derangement graph and use properties of the perfect matching polytope. We also prove that the perfect matching derangement graph is not a Cayley graph.
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…
We introduce uniformly vertex-transitive graphs as vertex-transitive graphs satisfying a stronger condition on their automorphism groups, motivated by a problem which arises from a Sinkhorn-type algorithm. We use the derangement graph $D(Γ)$ of a given graph $Γ$ to show that the uniform vertex-transitivity of $Γ$ is equivalent to the existence of cliques of sufficient size in $D(Γ)$. Using this method, we find examples of graphs that are vertex-transitive but not uniformly vertex-transitive, settling a previously open question. Furthermore, we develop sufficient criteria for uniform vertex-tra…
The signless Laplacian matrix of a graph $G$ is defined to be the sum of its adjacency matrix and degree diagonal matrix, and its eigenvalues are called $Q$-eigenvalues of $G$. A $Q$-eigenvalue of a graph $G$ is called a $Q$-main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. Chen and Huang [L. Chen, Q.X. Huang, Trees, unicyclic graphs and bicyclic graphs with exactly two $Q$-main eigenvalues, submitted for publication] characterized all trees, unicylic graphs and bicyclic graphs with exactly two main $Q$-eigenvalues, respectively. As a continuance of it, in…
In this paper, we show that the eigenvalues of certain classes of Cayley graphs are integers. The (n,k,r)-arrangement graph A(n,k,r) is a graph with all the k-permutations of an n-element set as vertices where two k-permutations are adjacent if they differ in exactly r positions. We establish a relation between the eigenvalues of the arrangement graphs and the eigenvalues of certain Cayley graphs. As a result, the conjecture on integrality of eigenvalues of A(n,k,1) follows.
A signless Laplacian eigenvalue of a graph $G$ is called a main signless Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, we first give the necessary and sufficient conditions for a graph with one main signless Laplacian eigenvalue or two main signless Laplacian eigenvalues, and then characterize the trees and unicyclic graphs with exactly two main signless Laplacian eigenvalues, respectively.
A signless Laplacian eigenvalue of a graph $G$ is called a main signless Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, all connected bicyclic graphs with exactly two main eigenvalues are determined.
Some recent results on graph eigenvalues are improved. In particular, among all graphs of given order with no cliques of order $(r+1)$ the $r$-partite Turan graph has maximal spectral radius.
Dom de Caen posed the question whether connected graphs with three distinct eigenvalues have at most three distinct valencies. We do not answer this question, but instead construct connected graphs with four and five distinct eigenvalues and arbitrarily many distinct valencies. The graphs with four distinct eigenvalues come from regular two-graphs. As a side result, we characterize the disconnected graphs and the graphs with three distinct eigenvalues in the switching class of a regular two-graph.
We give a survey on graphs with fixed smallest eigenvalue, especially on graphs with large minimal valency and also on graphs with good structures. Our survey mainly consists of the following two parts: (i) Hoffman graphs, the basic theory related to Hoffman graphs and the applications of Hoffman graphs to graphs with fixed smallest eigenvalue and large minimal valency; (ii) recent results on distance-regular graphs and co-edge regular graphs with fixed smallest eigenvalue and the characterizations of certain families of distance-regular graphs. At the end of the survey, we also discuss signed…
In this paper, we derive an upper bound for higher order eigenvalues of the normalized Laplace operator associated with a symmetric finite graph in terms of lower order eigenvalues.
We survey some of the known results on eigenvalues of Cayley graphs and their applications, together with related results on eigenvalues of Cayley digraphs and generalizations of Cayley graphs.
In this paper, we characterize all connected graphs with exactly three distinct normalized Laplacian eigenvalues of which one is equal to $1$, determine all connected bipartite graphs with at least one vertex of degree $1$ having exactly four distinct normalized Laplacian eigenvalues, and find all unicyclic graphs with three or four distinct normalized Laplacian eigenvalues.
Among all simple nonbipartite 2-connected graphs and among all nonbipartite $θ$-graphs, the minimum least $Q$-eigenvalues are completely determined, respectively.
A Neumaier graph is a non-complete edge-regular graph containing a regular clique. In this paper we give some sufficient and necessary conditions for a Neumaier graph to be strongly regular. Further we show that there does not exist Neumaier graphs with exactly four distinct eigenvalues. We also determine the Neumaier graphs with smallest eigenvalue -2.
In this paper, we introduce the concepts of the plain eigenvalue, the main-plain index and the refined spectrum of graphs. We focus on the graphs with two main and two plain eigenvalues and give some characterizations of them.
We give inequalities relating the eigenvalues of the adjacency matrix and the Laplacian of a graph, and its minimum and maximum degrees. The results are applied to derive new conditions for quasi-randomness of graphs.
In 1977 Smith characterized graphs with exactly one positive eigenvalue. Since then, many particular results related to graphs with exactly two positive eigenvalues have emerged. In this paper we conclude this investigation by giving a full characterization of these graphs.
We investigate properties of signed graphs that have few distinct eigenvalues together with a symmetric spectrum. Our main contribution is to determine all signed $(0,2)$-graphs with vertex degree at most $6$ that have precisely two distinct eigenvalues $\pm λ$. Next, we consider to what extent induced subgraphs of signed graph with two distinct eigenvalues $\pm λ$ are determined by their spectra. Lastly, we classify signed $(0,2)$-graphs that have a symmetric spectrum with three distinct eigenvalues and give a partial classification for those with four distinct eigenvalues.
This paper deals with the eigenvalues of the adjacency matrices of threshold graphs for which $-1$ and $0$ are considered as trivial eigenvalues. We show that threshold graphs have no non-trivial eigenvalues in the interval $\left[(-1-\sqrt{2})/2,\,(-1+\sqrt{2})/2\right]$. This confirms a conjecture by Aguilar, Lee, Piato, and Schweitzer (2018).
The classical problem of characterizing the graphs with bounded eigenvalues may date back to the work of Smith in 1970. Especially, the research on graphs with smallest eigenvalues not less than $-2$ has attracted widespread attention. Mixed graphs are natural generalization of undirected graphs. In this paper, we completely characterize the mixed graphs with smallest Hermitian eigenvalue greater than $-\frac{\sqrt{5}+1}{2}$, which consists of three infinite classes of mixed graphs and $30$ scattered mixed graphs. By the way, we get a new class of mixed graphs switching equivalent to their und…
We conjecture that the sum of the square roots of the Laplacian eigenvalues of a graph does not exceed the sum of the square roots of its signless Laplacian eigenvalues.
In this paper, we show that a connected graph with smallest eigenvalue at least -3 and large enough minimal degree is 2-integrable. This result generalizes a 1977 result of Hoffman for connected graphs with smallest eigenvalue at least -2.
