Distance-Regular Graphs with Strongly Regular Subconstituents

In [3] Cameron et al. classified strongly regular graphs with strongly regular subconstituents. Here we prove a theorem which implies that distance-regular graphs with strongly regular subconstituents are precisely the Taylor graphs and graphs with a ₁ = 0 and a _i {0,1} for i = 2,...,d.     

$c$-半层空间与几乎完全正则空间一点注记

In this paper, we give some characterizations of almost completely regular spaces and c-semistratifiable spaces(CSS) by semi-continuous functions. We mainly show that:(1)Let X be a space. Then the following statements are equivalent:(i) X is almost completely regular.(ii) Every two disjoint subsets of X, one of which is compact and the other is regular closed, are completely separated.(iii) If g, h : X → I, g is compact-like, h is normal lower semicontinuous, and g ≤ h, then there exists a continuous function f : X → I such that g ≤ f ≤ h;and(2) Let X be a space. Then the following statements are equivalent:(a) X is CSS;(b) There is an operator U assigning to a decreasing sequence of compact sets(Fj)j∈N,a decreasing sequence of open sets(U(n,(Fj)))n∈N such that(b1) Fn■U(n,(Fj)) for each n ∈ N;(b2)∩n∈NU(n,(Fj)) =∩n∈NFn;(b3) Given two decreasing sequences of compact sets(Fj)j∈N and(Ej)j∈N such that Fn■Enfor each n ∈ N, then U(n,(Fj))■U(n,(Ej)) for each n ∈ N;(c) There is an operator Φ : LCL(X, I) → USC(X, I) such that, for any h ∈ LCL(X, I),0 Φ(h) h, and 0 Φ(h)(x) h(x) whenever h(x) 0.     

简单完整正则平面图

对简单完整正则平面图的特性和结构进行了分析和讨论 ,找出了简单完整正则平面图的可能的种类 .此外 ,对各种简单完整正则平面图的色数进行了求解 ,并用不同的方法给出了各个简单完整正则平面图的作色方案 .     

Completely regular designs

We study a class of t-designs which enjoy a high degree of regularity. These are the subsets of vertices of the Johnson graph which are completely regular, in the sense of Delsarte [Philips Res. Reports Suppl. 10 (1973)]. After setting up the basic theory, we describe the known completely regular designs. We derive very strong restrictions which must hold in order for a design to be completely regular. As a result, we are able to determine which symmetric designs are completely regular and which Steiner systems with t = 2 are completely regular. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 261–273, 1998     

Uniformly Packed Codes and More Distance Regular Graphs from Crooked Functions

Let V and W be n-dimensional vector spaces over GF(2). A function Q : V W is called crooked (a notion introduced by Bending and Fon-Der-Flaass) if it satisfies the following three properties:

Abel正则球与完全正则半群

本文分别讨论了关于结合环和半群的二个定理，并且由结合环的这二个定理推出了如下准则：结合环Ｒ是Ａｂｅｌ正则的，当且仅当Ｒ的每个拟理想是正则环．     

Strongly Regular Decompositions of the Complete Graph

We study several questions about amorphic association schemes and other strongly regular decompositions of the complete graph. We investigate how two commuting edge-disjoint strongly regular graphs interact. We show that any decomposition of the complete graph into three strongly regular graphs must be an amorphic association scheme. Likewise we show that any decomposition of the complete graph into strongly regular graphs of (negative) Latin square type is an amorphic association scheme. We study strongly regular decompositions of the complete graph consisting of four graphs, and find a primitive counterexample to A.V. Ivanov's conjecture which states that any association scheme consisting of strongly regular graphs only must be amorphic.     

Globals of Completely Regular Monoids

An element of a semigroup S is called irreducible if it cannot be expressed as a product of two elements in S both distinct from itself. In this paper we show that the class C of all completely regular...     

On Perfect 2‐Colorings of Johnson Graphs J(v, 3)

We study the perfect 2‐colorings (also known as the equitable partitions into two parts or the completely regular codes with covering radius 1) of the Johnson graphs . In particular, we classify all the realizable quotient matrices of perfect 2‐colorings for odd v. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 21: 232–252, 2013     

Completely Regular Designs of Strength One

We study a class of highly regular t-designs. These are the subsets of vertices of the Johnson graph which are completely regular in the sense of Delsarte [2]. In [9], Meyerowitz classified the completely regular designs having strength zero. In this paper, we determine the completely regular designs having strength one and minimum distance at least two. The approach taken here utilizes the incidence matrix of (t+1)-sets versus k-sets and is related to the representation theory of distance-regular graphs [1, 5].     

幂等元生成子半群为完全正则半群的拟完全正则半群

论文主要刻画了幂等元生成子半群为完全正则半群的拟完全正则半群. 并讨论了满足该类半群的一些子类.     

关于Fuzzy完全正则半群

本文先引入Ｆｕｚｚｙ左（Ｆｕｚｚｙ右）正则半群的概念，进而讨论Ｆｕｚｚｙ左（Ｆｕｚｚｙ右）正则半群以及Ｆｕｚｚｙ完全正则半群中Ｆｕｚｚｙ理想的一些代数性质。     

Antimagic Labeling of Regular Graphs

A graph is antimagic if there is a one‐to‐one correspondence such that for any two vertices , . It is known that bipartite regular graphs are antimagic and nonbipartite regular graphs of odd degree at least three are antimagic. Whether all nonbipartite regular graphs of even degree are antimagic remained an open problem. In this article, we solve this problem and prove that all even degree regular graphs are antimagic.     

Perfect Matchings of Regular Bipartite Graphs

Let G be a regular bipartite graph and . We show that there exist perfect matchings of G containing both, an odd and an even number of edges from X if and only if the signed graph , that is a graph G with exactly the edges from X being negative, is not equivalent to . In fact, we prove that for a given signed regular bipartite graph with minimum signature, it is possible to find perfect matchings that contain exactly no negative edges or an arbitrary one preselected negative edge. Moreover, if the underlying graph is cubic, there exists a perfect matching with exactly two preselected negative edges. As an application of our results we show that each signed regular bipartite graph that contains an unbalanced circuit has a 2‐cycle‐cover such that each cycle contains an odd number of negative edges.     

Cyclotomy and Strongly Regular Graphs

We consider strongly regular graphs defined on a finite field by taking the union of some cyclotomic classes as difference set. Several new examples are found.     

The Terwilliger Algebra Associated with a Set of Vertices in a Distance-Regular Graph

Let Γ be a distance-regular graph of diameter D. Let X denote the vertex set of Γ and let Y be a nonempty subset of X. We define an algebra τ = τ(Y). This algebra is finite dimensional and semisimple. If Y consists of a single vertex then τ is the corresponding subconstituent algebra defined by P. Terwilliger. We investigate the irreducible τ-modules. We define endpoints and thin condition on irreducible τ-modules as a generalization of the case when Y consists of a single vertex. We determine when an irreducible module is thin. When the module is generated by the characteristic vector of Y, it is thin if and only if Y is a completely regular code of Γ. By considering a suitable subset Y, every irreducible τ(x)-module of endpoint i can be regarded as an irreducible τ(Y)-module of endpoint 0.This research was partially supported by the Grant-in-Aid for Scientific Research (No. 12640039), Japan Society of the Promotion of Science. A part of the research was done when the author was visiting the Ohio State University.     

具有完全正则自同态半群的分裂图

本文给出了具有完全正则自同态半群的分裂图的结构特征．其证明方法有望应用于其他图族自同态半群的正则性及完全正则性的研究．     

完全正则半群的Cayley表示

本文利用点集映射和等价关系得出了完全正则半群的Ｃａｙｌｅｙ表示，从而解决了１９８７年Ｍ．Ｐｅｔｒｉｃｈ提出的一个公开问题．作为特例，还得到完全单半群的ｃａｙｌｅｙ表示．     

Strongly Regular Semi-Cayley Graphs

We consider strongly regular graphs = (V, E) on an even number, say 2n, of vertices which admit an automorphism group G of order n which has two orbits on V. Such graphs will be called strongly regular semi-Cayley graphs. For instance, the Petersen graph, the Hoffman–Singleton graph, and the triangular graphs T(q) with q 5 mod 8 provide examples which cannot be obtained as Cayley graphs. We give a representation of strongly regular semi-Cayley graphs in terms of suitable triples of elements in the group ring Z G. By applying characters of G, this approach allows us to obtain interesting nonexistence results if G is Abelian, in particular, if G is cyclic. For instance, if G is cyclic and n is odd, then all examples must have parameters of the form 2n = 4s ² + 4s + 2, k = 2s ² + s, = s ² – 1, and = s ²; examples are known only for s = 1, 2, and 4 (together with a noncyclic example for s = 3). We also apply our results to obtain new conditions for the existence of strongly regular Cayley graphs on an even number of vertices when the underlying group H has an Abelian normal subgroup of index 2. In particular, we show the nonexistence of nontrivial strongly regular Cayley graphs over dihedral and generalized quaternion groups, as well as over two series of non-Abelian 2-groups. Up to now these have been the only general nonexistence results for strongly regular Cayley graphs over non-Abelian groups; only the first of these cases was previously known.     

图是圈的完全正矩阵

本文证明,图是围的完全正矩阵Ａ当比较矩阵Ｍ（Ａ）的行列式大于零时,恰有两个极小秩１分解,而当ｄｅｔＭ（Ａ）＝０时,恰有一个极小秩１分解．