离散数学教材中传递闭包的两个问题

设R和F是有限集合X上的两个关系.借助关系图的概念,本文确定了分别满足t(R∪F)=t(R)∪t(F),t(s(R))=s(t(R))的R和F的充分必要条件,解决了耿素云、曲婉玲和王捍贫2002年编著的《离散数学教程》第40页和44页中的两个遗留问题.     

Simple transitive 2-representations of left cell 2-subcategories of projective functors for star algebras

In this article, we study simple transitive 2-representations of certain 2-subcategories of the 2-category of projective functors over a star algebra. We show that in the simplest case, which is associated to the Dynkin type A₂, simple transitive 2-representations are classified by cell 2-representations. In the general case we conjecture that there exist many simple transitive 2-representations which are not cell 2-representations and provide some evidence for our conjecture.     

点传递图有向连通度的下界

本文研究点传递有向图与定向留连通度的下界，对达到此下界的Ｃｈｙｌｅｙ有向图与定向图进行了刻划。     

Transitive digraphs with more than one end

In this paper we investigate infinite, locally finite, connected, transitive digraphs with more than one end. For undirected graphs with these properties it has been shown that they are trees as soon as they are 2-arc transitive. In the case of digraphs the situation is much more involved. We show that these graphs can have both thick and thin ends, even if they are highly arc transitive. Hence they are far away from being ‘tree-like’. On the other hand all known examples of digraphs with more than one end are either highly arc transitive or at most 1-arc transitive. We conjecture that infinite, locally finite, connected, 2-arc transitive digraphs with more than one end are highly arc transitive and prove that this conjecture holds for digraphs with prime in- and out-degree and connected cuts.     

An infinite series of regular edge‐ but not vertex‐transitive graphs

Let n be an integer and q be a prime power. Then for any 3 ≤ n ≤ q?1, or n=2 and q odd, we construct a connected q‐regular edge‐but not vertex‐transitive graph of order 2qⁿ⁺¹. This graph is defined via a system of equations over the finite field of q elements. For n=2 and q=3, our graph is isomorphic to the Gray graph. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 249–258, 2002     

Intransitive Cartesian decompositions preserved by innately transitive permutation groups

A permutation group is innately transitive if it has a transitive minimal normal subgroup, which is referred to as a plinth. We study the class of finite, innately transitive permutation groups that can be embedded into wreath products in product action. This investigation is carried out by observing that such a wreath product preserves a natural Cartesian decomposition of the underlying set. Previously we classified the possible embeddings in the case where the innately transitive group projects onto a transitive subgroup of the top group. In this article we prove that the transitivity assumption we made in the previous paper was not too restrictive. Indeed, the image of the projection into the top group can only be intransitive when the finite simple group that is involved in the plinth comes from a small list. Even then, the innately transitive group can have at most three orbits on an invariant Cartesian decomposition. A consequence of this result is that if is an innately transitive subgroup of a wreath product in product action, then the natural projection of into the top group has at most two orbits.

     

Sup and Max Properties for the Numerical Radius of Operators in Banach Spaces

The article deals with the following problem: given a bounded linear operator A in a Banach space X, how can multiplication of A by an operator of norm one (contraction) affect the numerical radius of A? The approach used in this work is close to that employed by Vieira and Kubrusly in 2007 for their study concerning spectral radius. It turns out that this study is closely related to the study of V-operators conducted in 2005 by Khatskevich, Ostrovskii, and Shulman; the results of this article demonstrate that in certain cases the obtained property of an operator implies that it is a V-operator, while in some other cases the converse is true.     

Cores of Imprimitive Symmetric Graphs of Order a Product of Two Distinct Primes

A retract of a graph Γ is an induced subgraph Ψ of Γ such that there exists a homomorphism from Γ to Ψ whose restriction to Ψ is the identity map. A graph is a core if it has no nontrivial retracts. In general, the minimal retracts of a graph are cores and are unique up to isomorphism; they are called the core of the graph. A graph Γ is G‐symmetric if G is a subgroup of the automorphism group of Γ that is transitive on the vertex set and also transitive on the set of ordered pairs of adjacent vertices. If in addition the vertex set of Γ admits a nontrivial partition that is preserved by G, then Γ is an imprimitive G‐symmetric graph. In this paper cores of imprimitive symmetric graphs Γ of order a product of two distinct primes are studied. In many cases the core of Γ is determined completely. In other cases it is proved that either Γ is a core or its core is isomorphic to one of two graphs, and conditions on when each of these possibilities occurs is given.     

Classification of Vertex‐Transitive Cubic Partial Cubes

Partial cubes are graphs isometrically embeddable into hypercubes. In this article, it is proved that every cubic, vertex‐transitive partial cube is isomorphic to one of the following graphs: , for , the generalized Petersen graph G (10, 3), the cubic permutahedron, the truncated cuboctahedron, or the truncated icosidodecahedron. This classification is a generalization of results of Bre?ar et  al. (Eur J Combin 25 (2004), 55–64) on cubic mirror graphs; it includes all cubic, distance‐regular partial cubes (P. M. Weichsel, Discrete Math 109 (1992), 297–306), and presents a contribution to the classification of all cubic partial cubes.     

Bounds on sets with few distances

We derive a new estimate of the size of finite sets of points in metric spaces with few distances. The following applications are considered:

•

we improve the Ray-Chaudhuri-Wilson bound of the size of uniform intersecting families of subsets;

•

we refine the bound of Delsarte-Goethals-Seidel on the maximum size of spherical sets with few distances;

•

we prove a new bound on codes with few distances in the Hamming space, improving an earlier result of Delsarte.

We also find the size of maximal binary codes and maximal constant-weight codes of small length with 2 and 3 distances.