Small Transitive Families of Subspaces

A family of subspaces of a complex separable Hilbert space is transitive if every bounded operator which leaves each of its members invariant is scalar. This article surveys some results concerning transitive families of small cardinality, and adds some new ones. The minimum cardinality of a transitive family in finite dimensions (greater than 2) is 4. In infinite dimensions a transitive pair of linear manifolds exists but the minimum cardinality of a transitive family of dense operator ranges or norm-closed subspaces is not known. However, a transitive family of dense operator ranges with 5 elements can be found, and so can a transitive family of norm-closed subspaces with 4 elements. In finite dimensions (> 1) three nest algebras (corresponding to maximal nests) can intersect in the scalar operators, but two cannot. It is not known if this is the case in infinite dimensions for maximal nests of type ω + 1. Four such nest algebras can intersect in the scalar operators. Received June 15, 2002, Accepted November 27, 2002     

Weighted external difference families and R-optimal AMD codes

In this paper, we provide a mathematical framework for characterizing AMD codes that are R-optimal. We introduce a new combinatorial object, the reciprocally-weighted external difference family (RWEDF), which corresponds precisely to an R-optimal weak AMD code. This definition subsumes known examples of existing optimal codes, and also encompasses combinatorial objects not covered by previous definitions in the literature. By developing structural group-theoretic characterizations, we exhibit infinite families of new RWEDFs, and new construction methods for known objects such as near-complete EDFs. Examples of RWEDFs in non-abelian groups are also discussed.     

An infinite family of sharply two-arc transitive digraphs

We disclaim Conjecture 1 posed by Seifter in [N. Seifter, Transitive digraphs with more than one end, Discrete Math., to appear], that stated that a connected locally finite digraph with more than one end is either 0-, 1- or highly arc transitive. We describe in this work an infinite family of 2-arc transitive two-ended digraphs, that are not 3-arc transitive.     

Neighbour transitivity on codes in Hamming graphs

We consider a code to be a subset of the vertex set of a Hamming graph. In this setting a neighbour of the code is a vertex which differs in exactly one entry from some codeword. This paper examines codes with the property that some group of automorphisms acts transitively on the set of neighbours of the code. We call these codes neighbour transitive. We obtain sufficient conditions for a neighbour transitive group to fix the code setwise. Moreover, we construct an infinite family of neighbour transitive codes, with minimum distance δ = 4, where this is not the case. That is to say, knowledge of even the complete set of code neighbours does not determine the code.     

On normal 2-geodesic transitive Cayley graphs

We investigate connected normal 2-geodesic transitive Cayley graphs Cay(T,S). We first prove that if Cay(T,S) is neither cyclic nor K_4[2], then 〈a〉?{1}??S for all a∈S. Next, as an application, we give a reduction theorem proving that each graph in this family which is neither a complete multipartite graph nor a bipartite 2-arc transitive graph, has a normal quotient that is either a complete graph or a Cayley graph in the family for a characteristically simple group. Finally we classify complete multipartite graphs in the family.     

Planar embeddings with infinite faces

We investigate vertex‐transitive graphs that admit planar embeddings having infinite faces, i.e., faces whose boundary is a double ray. In the case of graphs with connectivity exactly 2, we present examples wherein no face is finite. In particular, the planar embeddings of the Cartesian product of the r‐valent tree with K₂ are comprehensively studied and enumerated, as are the automorphisms of the resulting maps, and it is shown for r = 3 that no vertex‐transitive group of graph automorphisms is extendable to a group of homeomorphisms of the plane. We present all known families of infinite, locally finite, vertex‐transitive graphs of connectivity 3 and an infinite family of 4‐connected graphs that admit planar embeddings wherein each vertex is incident with an infinite face. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 257–275, 2003     

Unitary graphs and classification of a family of symmetric graphs with complete quotients

A finite graph Γ is called G-symmetric if G is a group of automorphisms of Γ which is transitive on the set of ordered pairs of adjacent vertices of Γ. We study a family of symmetric graphs, called the unitary graphs, whose vertices are flags of the Hermitian unital and whose adjacency relations are determined by certain elements of the underlying finite fields. Such graphs admit the unitary groups as groups of automorphisms, and play a significant role in the classification of a family of symmetric graphs with complete quotients such that an associated incidence structure is a doubly point-transitive linear space. We give this classification in the paper and also investigate combinatorial properties of the unitary graphs.     

The vertex-transitive TLF-planar graphs

We consider the class of the topologically locally finite (in short TLF) planar vertex-transitive graphs. We characterize these graphs by finite combinatorial objects called labeling schemes. As a result, we are able to enumerate and describe all TLF-planar vertex-transitive graphs of given degree, as well as most of their transitive groups of automorphisms. In addition,we are able to decide whether a given TLF-planar transitive graph is Cayley or not. This class contains all the one-ended planar Cayley graphs and the normal transitive tilings of the plane.     

On an extension of a combinatorial identity

Using Frobenius partitions we extend the main results of [4]. This leads to an infinite family of 4-way combinatorial identities. In some particular cases we get even 5-way combinatorial identities which give us four new combinatorial versions of Göllnitz-Gordon identities.     

Extension of Multiply Transitive Permutation Sets

Let G be a k-transitive permutation set on E and let E* = E∪{∞},∞ ? E; if G* is a (k: + 1)-transitive permutation set on E*, G* is said to be an extension of G whenever G _∝ ^* =G. In this work we deal with the problem of extending (sharply) k- transitive permutation sets into (sharply) (k + 1)-transitive permutation sets. In particular we give sufficient conditions for the extension of such sets; these conditions can be reduced to a unique one (which is a necessary condition too) whenever the considered set is a group. Furthermore we establish necessary and sufficient conditions for a sharply k- transitive permutation set (k ≥ 3) to be a group. Math. Subj. Class.: 20B20 Multiply finite transitive permutation groups 20B22 Multiply infinite transitive permutation groups     

A construction of an infinite family of 2-arc transitive polygonal graphs of arbitrary odd girth

A near-polygonal graph is a graph Γ which has a set C of m-cycles for some positive integer m such that each 2-path of Γ is contained in exactly one cycle in C. If m is the girth of Γ then the graph is called polygonal. We provide a construction of an infinite family of polygonal graphs of arbitrary odd girth with 2-arc transitive automorphism groups.     

Neighborly combinatorial 3-manifolds with dihedral automorphism group

It is well known that for anyn≧5 the boundary complex of the cyclic 4-polytopeC(n, 4) is a neighborly combinatorial 3-sphere admitting a vertex transitive action of the dihedral groupD _n of order 2n. In this paper we present a similar series of neighborly combinatorial 3-manifolds withn≧9 vertices, each homeomorphic to the “3-dimensional Klein bottle”. Forn=9 andn=10 these examples have been observed. before by A. Altshuler and L. Steinberg. Moreover we give a computer-aided enumeration of all neighborly combinatorial 3-manifolds with such a symmetry and with at most 19 vertices. It turns out that there are only four other types, one with 10, 15, 17, 19 vertices. We also discuss the more general case of manifolds with cyclic automorphism groupC _n.     

Intersections of Magnus subgroups and embedding theorems for cyclically presented groups

It is now known that the intersection of two Magnus subgroups M_i=〈Y_i〉 (1≤i≤2) in a one-relator group is either the free group F on Y₁∩Y₂ or the free product of F together with an infinite cyclic group (so-called exceptional intersection). Using this, we give conditions under which two embedding theorems for cyclically presented groups can be obtained. This provides a new method for proving such groups infinite. We also give a combinatorial method for checking the presence of exceptional intersections.     

Locally s-distance transitive graphs and pairwise transitive designs

The study of locally s-distance transitive graphs initiated by the authors in previous work, identified that graphs with a star quotient are of particular interest. This paper shows that the study of locally s-distance transitive graphs with a star quotient is equivalent to the study of a particular family of designs with strong symmetry properties that we call nicely affine and pairwise transitive. We show that a group acting regularly on the points of such a design must be abelian and give general construction for this case.     

Transitivity degrees of countable groups and acylindrical hyperbolicity

We prove that every countable acylindrically hyperbolic group admits a highly transitive action with finite kernel. This theorem uniformly generalizes many previously known results and allows us to answer a question of Garion and Glassner on the existence of highly transitive faithful actions of mapping class groups. It also implies that in various geometric and algebraic settings, the transitivity degree of an infinite group can only take two values, namely 1 and ∞. Here, by transitivity degree of a group we mean the supremum of transitivity degrees of its faithful permutation representations. Further, for any countable group G admitting a highly transitive faithful action, we prove the following dichotomy: Either G contains a normal subgroup isomorphic to the infinite alternating group or G resembles a free product from the model theoretic point of view. We apply this theorem to obtain new results about universal theory and mixed identities of acylindrically hyperbolic groups. Finally, we discuss some open problems.     

On tree‐decompositions of one‐ended graphs

A graph is one‐ended if it contains a ray (a one way infinite path) and whenever we remove a finite number of vertices from the graph then what remains has only one component which contains rays. A vertex v dominates a ray in the end if there are infinitely many paths connecting v to the ray such that any two of these paths have only the vertex v in common. We prove that if a one‐ended graph contains no ray which is dominated by a vertex and no infinite family of pairwise disjoint rays, then it has a tree‐decomposition such that the decomposition tree is one‐ended and the tree‐decomposition is invariant under the group of automorphisms. This can be applied to prove a conjecture of Halin from 2000 that the automorphism group of such a graph cannot be countably infinite and solves a recent problem of Boutin and Imrich. Furthermore, it implies that every transitive one‐ended graph contains an infinite family of pairwise disjoint rays.     

Infinite families of strange partition congruences for broken 2-diamonds

In 2007, George E. Andrews and Peter Paule (Acta Arithmetica 126:281–294, 2007) introduced a new class of combinatorial objects called broken k-diamonds. Their generating functions connect to modular forms and give rise to a variety of partition congruences. In 2008, Song Heng Chan proved the first infinite family of congruences when k=2. In this note, we present two non-standard infinite families of broken 2-diamond congruences derived from work of Oliver Atkin and Morris Newman. In addition, four conjectures related to k=3 and k=5 are stated.     

On non-antipodal binary completely regular codes

Binary non-antipodal completely regular codes are characterized. Using a result on nonexistence of nontrivial binary perfect codes, it is concluded that there are no unknown nontrivial non-antipodal completely regular binary codes with minimum distance d?3. The only such codes are halves and punctured halves of known binary perfect codes. Thus, new such codes with covering radius ρ=6 and 7 are obtained. In particular, a half of the binary Golay [23,12,7]-code is a new binary completely regular code with minimum distance d=8 and covering radius ρ=7. The punctured half of the Golay code is a new completely regular code with minimum distance d=7 and covering radius ρ=6. The new code with d=8 disproves the known conjecture of Neumaier, that the extended binary Golay [24,12,8]-code is the only binary completely regular code with d?8. Halves of binary perfect codes with Hamming parameters also provide an infinite family of binary completely regular codes with d=4 and ρ=3. Puncturing of these codes also provide an infinite family of binary completely regular codes with d=3 and ρ=2. Both these families of codes are well known, since they are uniformly packed in the narrow sense, or extended such codes. Some of these completely regular codes are new completely transitive codes.