Vertex-transitive Digraphs Research Articles

On weak metric dimension of digraphs

Using the two-way distance, we introduce the concept of the weak metric dimension of a strongly connected digraph [Formula: see text]. We first establish lower and upper bounds for the number of arcs in [Formula: see text] by using the diameter and weak metric dimension of [Formula: see text], and characterize all digraphs attaining the lower or upper bound. Then we study a digraph with weak metric dimension 1 and classify all vertex-transitive digraphs having weak metric dimension 1. Finally, all digraphs of order n with weak metric dimension [Formula: see text] or [Formula: see text] are determined.

Arc-transitive graphs of valency twice a prime admit a semiregular automorphism

We prove that every finite arc-transitive graph of valency twice a prime admits a nontrivial semiregular automorphism, that is, a non-identity automorphism whose cycles all have the same length. This is a special case of the Polycirculant Conjecture, which states that all finite vertex-transitive digraphs admit such automorphisms.

On the classification of vertex-transitive structures

We consider the classification problem for several classes of countable structures which are "vertex-transitive", meaning that the automorphism group acts transitively on the elements. (This is sometimes called homogeneous.) We show that the classification of countable vertex-transitive digraphs and partial orders are Borel complete. We identify the complexity of the classification of countable vertex-transitive linear orders. Finally we show that the classification of vertex-transitive countable tournaments is properly above $E_0$ in complexity.

Game of cops and robbers in oriented quotients of the integer grid

The integer grid Z□Z has four typical orientations of its edges which make it a vertex-transitive digraph. In this paper we analyze the game of Cops and Robbers on arbitrary finite quotients of these directed grids.

Connectivity of half vertex transitive digraphs

A bipartite digraph is said to be a half vertex transitive digraph if its automorphism acts transitively on the sets of its bipartition, respectively. In this paper, bipartite double coset digraphs of groups are defined and it is shown that any half vertex transitive digraph is isomorphic to some half double coset digraph, and we show that the connectivity of any strongly connected half transitive digraph is its minimum degree.

Vertex-Transitive Digraphs of Order $p^5$ are Hamiltonian

We prove that connected vertex-transitive digraphs of order $p^{5}$ (where $p$ is a prime) are Hamiltonian, and a connected digraph whose automorphism group contains a finite vertex-transitive subgroup $G$ of prime power order such that $G'$ is generated by two elements or elementary abelian is Hamiltonian.

Semiregular automorphisms of edge-transitive graphs

The polycirculant conjecture asserts that every vertex-transitive digraph has a semiregular automorphism: a non-trivial automorphism whose cycles all have the same length. In this paper, we investigate the existence of semiregular automorphisms of edge-transitive graphs. In particular, we show that any regular edge-transitive graph of valency three or four has a semiregular automorphism.

Automorphism group of the Comellas-Fiol digraphs

In 1995, Comellas and Fiol proposed a construction of record large vertex-transitive digraphs of given degree and diameter out of relatively small vertex-transitive input digraphs. We determine the automorphism group of the output digraph for a special but rather wide class of input digraphs.

On Vosperian and Superconnected Vertex-Transitive Digraphs

We investigate the structure of a digraph having a transitive automorphism group where every cutset of minimal cardinality consists of all successors or all predecessors of some vertex. We give a complete characterization of vosperian arc-transitive digraphs. It states that an arc-transitive strongly connected digraph is vosperian if and only if it is irreducible. In particular, this is the case if the degree is coprime with the order of the digraph. We give also a complete characterization of vosperian Cayley digraphs and a complete characterization of irreducible superconnected Cayley digraphs. These two last characterizations extend the corresponding ones in Abelian Cayley digraphs and the ones in the undirected case.

Which Faber–Moore–Chen digraphs are Cayley digraphs?

The problem of finding the largest graphs and digraphs of given degree and diameter is known as the ‘degree–diameter’ problem. One of the families of largest known vertex-transitive digraphs of given degree and diameter is the Faber–Moore–Chen digraphs. In our contribution we will classify those Faber–Moore–Chen digraphs that are Cayley digraphs.

Semiregular automorphisms of arc-transitive graphs with valency pq

It was conjectured (see [D. Marušič, On vertex symmetric digraphs, Discrete Math. 36 (1981) 69–81]) that every vertex-transitive digraph has a semiregular automorphism, that is, a nonidentity automorphism having all orbits of equal length. Despite several partial results supporting its content, the conjecture remains open. In this paper, it is shown that the conjecture holds whenever the graph is arc-transitive of valency p q , where p and q are primes ( p may equal q ), and such that its automorphism group has a nonabelian minimal normal subgroup with at least three orbits on the vertex set.

Distance Sequences In Locally Infinite Vertex-Transitive Digraphs

We prove that the out-distance sequence {f+(k)} of a vertex-transitive digraph of finite or infinite degree satisfies f+(k+1)≤f+(k)2 for k≥1, where f+(k) denotes the number of vertices at directed distance k from a given vertex. As a corollary, we prove that for a connected vertex-transitive undirected graph of infinite degree d, we have f(k)=d for all k, 1≤k<diam(G). This answers a question by L. Babai.

All vertex-transitive locally-quasiprimitive graphs have a semiregular automorphism

The polycirculant conjecture states that every transitive 2-closed permutation group of degree at least two contains a nonidentity semiregular element, that is, a nontrivial permutation whose cycles all have the same length. This would imply that every vertex-transitive digraph with at least two vertices has a nonidentity semiregular automorphism. In this paper we make substantial progress on the polycirculant conjecture by proving that every vertex-transitive, locally-quasiprimitive graph has a nonidentity semiregular automorphism. The main ingredient of the proof is the determination of all biquasiprimitive permutation groups with no nontrivial semiregular elements.

On the change of the Jordan form under the transition from the adjacency matrix of a vertex-transitive digraph to its principal submatrix of co-order one

Let J( λ; n 1, …, n k ) be the set of matrices A such that λ is an eigenvalue of A and n 1 ⩽ ⋯ ⩽ n k are the sizes of the Jordan blocks associated with λ. For a given index v of A, denote by A − v the principal submatrix of co-order one obtained from A by deleting the vth row and column. In the present paper, all possible changes of the part of the Jordan form corresponding to λ under the transition from A to A − v are determined for matrices A ∈ J( λ; n 1, …, n k ) such that for the eigenvalue λ of both A and A ⊤, there exists a Jordan chain of the largest length n k whose eigenvector has nonzero vth entry. In particular, it is shown that for almost every matrix A ∈ J( λ; n 1, …, n k ), n 1, …, n k−1 are the sizes of Jordan blocks for λ considered as an eigenvalue of A − v. Moreover, it is also proved that if A is the adjacency matrix of a vertex-transitive digraph and k ⩾ 2, then the change n 1, …, n k → n 1, …, n k−2 , 2 n k−1 − 1 holds for the eigenvalue λ under the transition from A to A − v. In the case of k = 1, λ is a simple eigenvalue of A and does not belong to the spectrum of A − v.

Infinite families of non-Cayley vertex-transitive tournaments

In this work is proven the existence of non-Cayley vertex-transitive tournaments of order p k , for each prime p ⩾ 3 and k ⩾ 4 , and an explicit construction is given. These tournaments are special cases of ( p k - 1 , p ) -metacirculant digraphs, and have the same automorphism group as the first non-Cayley vertex-transitive digraphs of order p k , given by Marušič in 1985. Moreover, from these tournaments, new non-Cayley vertex-transitive tournaments that realise many more degrees are constructed.

Permutation Groups, Vertex-transitive Digraphs and Semiregular Automorphisms

A nonidentity element of a permutation group is said to be semiregular if all of its orbits have the same length. The work in this paper is linked to [6] where the problem of existence of semiregular automorphisms in vertex-transitive digraphs was posed. It was observed there that every vertex-transitive digraph of orderpkormp, wherepis a prime,k≥1 andm≤pare positive integers, has a semiregular automorphism. On the other hand, there are transitive permutation groups without semiregular elements [4]. In this paper, it is proved that every cubic vertex-transitive graph contains a semiregular automorphism, and moreover, it is shown that every vertex-transitive digraph of order 2p2, wherepis a prime, contains a semiregular automorphism.

Some general aspects of the framing number of a digraph

A digraph D is homogeneously embedded in a digraph H if for each vertex x of D and each vertex y of H, there exists an embedding of D in H as an induced subdigraph with x at y. A digraph F of minimum order in which D can be homogeneously embedded is called a frame of D and the order of F is called the framing number of D. Several general results involving frames and framing numbers of digraphs are established. The framing number is determined for a number of classes of digraphs, including a class of digraphs whose underlying graph is a complete bipartite graph, a class of digraphs whose underlying graph is C n + K 1, and the lexicographic product of a transitive tournament and a vertex transitive digraph. A relationship between the diameters of the underlying graphs of a digraph and its frame is determined. We show that every tournament has a frame which is also a tournament.

Digraphs on permutations

We study a family of vertex transitive digraphs whose vertices represent the k-permutations of n elements. After showing some general properties, we concentrate upon the study of the symmetry of these digraphs. By using some distance-related properties, their automorphism groups are characterized. We also characterize those digraphs which are Cayley digraphs. Finally, the diameter of these digraphs is obtained for values of n and k which include almost all values for which they are Cayley digraphs.

Determining the hamilton-connectedness of certain vertex-transitive graphs

Faber and Moore have proposed a class of vertex-transitive digraphs as a model of directed inconnection networks. These networks have attractive degree versus diameter properties. We show that these digraphs are Hamiltonian and provide necessary and sufficient conditions for the existence of a Hamilton path between any two vertices. Particular cases of these digraphs are the right rotation Cayley digraphs Cay( X n : S n where X n is the set of right rotations (1 2 s ̇ i −1 i) for i=2,3,…,n . Another class of vertex-transitive graphs that have been proposed as models of inconnection networks are the arrangement graphs introduced by Day and Tripathi. They have the same vertex set as the Faber–Moore digraphs. We prove that arrangement graphs are Hamilton-connected.

A Lloyd Theorem in Weakly Metric Association Schemes

Weakly metric association schemes are a generalization of metric schemes occurring in graph theory (vertex transitive digraphs), coding theory (Lee scheme, Clark-Liang scheme), and group theory (conjugacy scheme). Linear constraints on the inner distribution of r-covering codes in these schemes lead to a ‘Lloyd theorem’ on perfect codes. This generalizes results from Delsarte, Biggs, Bassalygo and Landauer.