Search

In 1992, Terwilliger introduced the notion of the Terwilliger algebra in order to study association schemes. The Terwilliger algebra of an association scheme A is the subalgebra of the complex matrix algebra, generated by the Bose-Mesner algebra of A and its dual idempotents with respect to a point x. In Bannai and Munemasa (1995) [3] determined the dimension of the Terwilliger algebra of abelian groups and dihedral groups, by showing that they are triply transitive (i.e., triply regular and dually triply regular). In this paper, we give a generalization of their results to the group association scheme of semidirect products of the form C n ⋊ C 2 , where C m is a cyclic group of order m ≥ 2. Moreover, we will give the complete characterization of the Wedderburn components of the Terwilliger algebra of these groups. [ABSTRACT FROM AUTHOR]

Introduced by Solomon in his 1976 paper, the descent algebra of a finite Coxeter group received significant attention over the past decades. As proved by Gessel, in the case of the symmetric group its structure constants give the comultiplication table for the fundamental basis of quasisymmetric functions. We show that this latter property actually implies several well known relations linked to the Robinson–Schensted–Knuth correspondence and some of its generalisations. This provides a new link between these results and the theory of quasisymmetric functions and allows to derive more advanced formulae involving Kronecker coefficients. Using the theory of type B quasisymmetric functions introduced by Chow, we extend this connection to the hyperoctahedral group and derive new formulae relating the structure constants of the descent algebra of type B, the numbers of domino tableaux of given descent set and the Kronecker coefficients of the hyperoctahedral group. [ABSTRACT FROM AUTHOR]

Abstract: A nonidentity automorphism of a graph is said to be semiregular if all of its orbits are of the same length. Given a graph with a semiregular automorphism , the quotient of relative to is the multigraph whose vertices are the orbits of and two vertices are adjacent by an edge with multiplicity if every vertex of one orbit is adjacent to vertices of the other orbit. We say that is an expansion of . In [J.D. Horton, I.Z. Bouwer, Symmetric -graphs and -graphs, J. Combin. Theory Ser. B 53 (1991) 114–129], Horton and Bouwer considered a restricted sort of expansions (which we will call ‘strong’ in this paper) where every leaf of expands to a single cycle in . They determined all cubic arc-transitive strong expansions of simple (1, 3)-trees, that is, trees with all of their vertices having valency 1 or 3, thus extending the classical result of Frucht, Graver and Watkins (see [R. Frucht, J.E. Graver, M.E. Watkins, The groups of the generalized Petersen graphs, Proc. Cambridge Philos. Soc. 70 (1971) 211–218]) about arc-transitive strong expansions of (also known as the generalized Petersen graphs). In this paper another step is taken further by considering the possible structure of cubic vertex-transitive expansions of general (1,3)-multitrees (where vertices with double edges are also allowed); thus the restriction on every leaf to be expanded to a single cycle is dropped. [Copyright &y& Elsevier]

Abstract: This paper investigates a competitive version of the coloring game on a finite graph . An asymmetric variant of the -relaxed coloring game is called the -relaxed -coloring game. In this game, two players, Alice and Bob, take turns coloring the vertices of a graph , using colors from a set , with . On each turn Alice colors vertices and Bob colors vertices. A color is legal for an uncolored vertex if by coloring with color , the subgraph induced by all the vertices colored with has maximum degree at most . Each player is required to color an uncolored vertex legally on each move. The game ends when there are no remaining uncolored vertices. Alice wins the game if all vertices of the graph are legally colored, Bob wins if at a certain stage there exists an uncolored vertex without a legal color. The -relaxed -game chromatic number, denoted by , of is the least for which Alice has a winning strategy in the -relaxed -coloring game. The -relaxed -coloring game has been well studied and there are many interesting results. For the -relaxed -coloring game, this paper proves that if a graph has an orientation with maximum outdegree and , then for all ; If , then - for all . [Copyright &y& Elsevier]

Abstract: Partial cubes are isometric subgraphs of hypercubes. Structures on a graph defined by means of semicubes, and Djoković’s and Winkler’s relations play an important role in the theory of partial cubes. These structures are employed in the paper to characterize bipartite graphs and partial cubes of arbitrary dimension. New characterizations are established and new proofs of some known results are given. The operations of Cartesian product and pasting, and expansion and contraction processes are utilized in the paper to construct new partial cubes from old ones. In particular, the isometric and lattice dimensions of finite partial cubes obtained by means of these operations are calculated. [Copyright &y& Elsevier]

Abstract: This paper discusses the game colouring number of partial k-trees and planar graphs. Let and denote the maximum game colouring number of partial k trees and the maximum game colouring number of planar graphs, respectively. In this paper, we prove that and . We also prove that the game colouring number of a graph is a monotone parameter, i.e., if is a subgraph of , then . [Copyright &y& Elsevier]

Abstract: Let us call a digraph D cycle-connected if for every pair of vertices there exists a cycle containing both u and . In this paper we study the following open problem introduced by Ádám. Let D be a cycle-connected digraph. Does there exist a universal edge in D, i.e., an edge such that for every there exists a cycle C such that and ? In his 2001 paper Hetyei conjectured that cycle-connectivity always implies the existence of a universal edge. In the present paper we prove the conjecture of Hetyei for bitournaments. [Copyright &y& Elsevier]

Abstract: This paper deals with multialgebras. An important tool in the theory of multialgebras is the fundamental relation, which brings us into the class of the universal algebras. In this paper we will present the construction of the direct limit of a direct system of multialgebras, some basic properties of this construction, and we will see that the fundamental algebra of the direct limit of multialgebras is the direct limit of the fundamental algebras of the given multialgebras. [Copyright &y& Elsevier]

Abstract: For a digraph D, let and denote its line digraph and subdivision digraph, respectively. The motivation of this paper is to solve the digraph equation . We show that and are cospectral if and only if D and have the same number of arcs. Further, we characterize the situation that and are isomorphic. Our approach introduces the new notion, the proper image of a digraph , and a new type of connectedness for digraphs. The concept plays an important role in the main result of this paper. It is also useful in other aspects of the study of line digraphs. For example, is connected if and only if is connected; is functional (contrafunctional) if and only if is functional (contrafunctional). Some related results are also presented. [Copyright &y& Elsevier]

Abstract: A graph G is said to be bicritical if has a perfect matching for every choice of a pair of points and . Bicritical graphs play a central role in decomposition theory of elementary graphs with respect to perfect matchings. As Plummer pointed out many times, the structure of bicritical graphs is far from completely understood. This paper presents a concise structure characterization on bicritical graphs in terms of factor-critical graphs and transversals of hypergraphs. A connected graph G with at least points is said to be k-extendable if it contains a matching of k lines and every such matching is contained in a perfect matching. A structure characterization for k-extendable bipartite graphs is given in a recursive way. Furthermore, this paper presents an algorithm for determining the extendability of a bipartite graph G, the maximum integer k such that G is k-extendable, where n is the number of points and m is the number of lines in G. [Copyright &y& Elsevier]

Abstract: In this paper we formulate four families of problems with which we aim at distinguishing different levels of randomness. The first one is completely non-random, being the ordinary Ramsey–Turán problem and in the subsequent three problems we formulate some randomized variations of it. As we will show, these four levels form a hierarchy. In a continuation of this paper we shall prove some further theorems and discuss some further, related problems. [Copyright &y& Elsevier]

Abstract: Demetrovics et al [Design type problems motivated by database theory, J. Statist. Plann. Inference 72 (1998) 149–164] constructed a decomposition of the family of all k-element subsets of an n-element set into disjoint pairs where two such pairs are relatively far from each other in some sense. The paper invented a proof method using a Hamiltonian-type theorem. The present paper gives a generalization of this tool, hopefully extending the power of the method. Problems where the method could be also used are shown. Moreover, open problems are listed which are related to the Hamiltonian theory. In these problems a cyclic permutation is to be found when certain restrictions are given by a family of k-element subsets. [Copyright &y& Elsevier]

Abstract: This paper is devoted to the lower bounds on the maximum genus of graphs. A simple statement of our results in this paper can be expressed in the following form: Let G be a k-edge-connected graph with minimum degree , for each positive integer , there exists a non-decreasing function such that the maximum genus of G satisfies the relation , and furthermore that , where is the cycle rank of G. The result shows that lower bounds of the maximum genus of graphs with any given connectivity become larger and larger as their minimum degree increases, and complements recent results of several authors. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to define the notions of -valued Lukasiewicz-Moisil algebra of fractions and maximal algebra of fractions taking as a guide-line the elegant construction of a complete ring of fractions by partial morphisms introduced by [Lambek, 1996. Lectures on Rings and Modules, p. 36]. For some informal explanations of the notion of fraction see [Lambek, 1996. Lectures on Rings and Modules, p. 36] In the last part of this paper we prove the existence of a maximal n-valued Lukasiewicz–Moisil algebra of fractions for an n-valued Lukasiewicz–Moisil algebra (Theorem 3.2) and we give an explicit description of this -valued Lukasiewicz–Moisil algebra for some classes of -valued Lukasiewicz–Moisil algebras (finite, chains, Boolean algebras). [Copyright &y& Elsevier]

Abstract: We consider the problem of reconstructing a partition x of the integer n from the set of its t-subpartitions. These are the partitions of the integer obtained by deleting a total of t from the parts of x in all possible ways. It was shown (in a forthcoming paper) that all partitions of n can be reconstructed from t-subpartitions if n is sufficiently large in relation to t. In this paper we deal with efficient reconstruction, in the following sense: if all partitions of n are -reconstructible, what is the minimum number such that every partition of n can be identified from any distinct subpartitions? We determine the function and describe the corresponding algorithm for reconstruction. Superpartitions may be defined in a similar fashion and we determine also the maximum number of t-superpartitions common to two distinct partitions of n. [Copyright &y& Elsevier]

Abstract: In the present paper, we consider the statistic “number of udu''s” in Dyck paths. The enumeration of Dyck paths according to semilength and various other parameters has been studied in several papers. However, the statistic “number of udu''s” has been considered only recently. Let denote the set of Dyck paths of semilength n and let , and denote the number of Dyck paths in with k udu''s, with k udu''s at low level, at high level, and at level , respectively. We derive their generating functions, their recurrence relations and their explicit formulas. A new setting counted by Motzkin numbers is also obtained. Several combinatorial identities are given and other identities are conjectured. [Copyright &y& Elsevier]

The initial point of this paper are two Kruskal–Katona type theorems. The first is the Colored Kruskal–Katona Theorem which can be stated as follows: Direct products of the form Bk11×Bk21×⋯×Bkn1 belong to the class of Macaulay posets, where Bkt denotes the poset consisting of the t+1 lowest levels of the Boolean lattice Bk. The second one is a recent result saying that also the products Bk1k1−1×Bk2k2−1×⋯×Bknkn−1 are Macaulay posets. The main result of this paper is that the natural common generalization to products of truncated Boolean lattices does not hold, i.e. that (Bkt)n is a Macaulay poset only if t∈{0,1,k−1,k}. [Copyright &y& Elsevier]

Nous poursuivons ici le travail commencé dans (Enjalbert and Hoang Ngoc Minh, 2012) en décrivant des produits de mélanges d’algèbres de plus en plus “grandes” de fonctions spéciales (issues d’équations différentielles à pôles simples). Les étudier nous conduit à définir une classe de produits de mélange, que nous nommons φ -shuffles. Nous étudions cette classe d’un point de vue combinatoire, en commençant par étendre (sous conditions) le théorème de Radford à celle-ci, puis en construisant (toujours sous conditions) sa bigèbre. Nous analysons les conditions des résultats précités pour les simplifier en les rendant visible dès la définition du produit de mélange. Nous testons enfin ces conditions sur les produits introduits en début d’article. We carry on the investigation initiated in Enjalbert and Hoang Ngoc Minh (2012): we describe new shuffle products coming from some special functions and group them, along with other products encountered in the literature, in larger and larger classes of products, which we name φ -shuffle products. Our paper is dedicated to a study of the latter class, from a combinatorial standpoint. We consider first how to extend Radford’s theorem to the products in that class, then how to construct their bi-algebras. As some conditions are necessary to carry that out, we study them closely and simplify them so that they can be seen directly from the definition of the product. We eventually test these conditions on the products mentioned above. [ABSTRACT FROM AUTHOR]

Let Γ denote a Q -polynomial distance-regular graph with vertex set X and diameter D. Let A denote the adjacency matrix of Γ. For a vertex x ∈ X and for 0 ≤ i ≤ D , let E i ⁎ (x) denote the projection matrix to the i th subconstituent space of Γ with respect to x. The Terwilliger algebra T (x) of Γ with respect to x is the semisimple subalgebra of Mat X (C) generated by A , E 0 ⁎ (x) , E 1 ⁎ (x) , ... , E D ⁎ (x). Let V denote a C -vector space consisting of complex column vectors with rows indexed by X. We say Γ is pseudo-vertex-transitive whenever for any vertices x , y ∈ X , there exists a C -vector space isomorphism ρ : V → V such that (ρ A − A ρ) V = 0 and (ρ E i ⁎ (x) − E i ⁎ (y) ρ) V = 0 for all 0 ≤ i ≤ D. In this paper, we discuss pseudo-vertex transitivity for distance-regular graphs with diameter D ∈ { 2 , 3 , 4 }. For D = 2 , we show that a strongly regular graph is pseudo-vertex-transitive if and only if all its local graphs have the same spectrum. For D = 3 , we consider the Taylor graphs and show that they are pseudo-vertex transitive. For D = 4 , we consider the antipodal tight graphs and show that they are pseudo-vertex transitive. [ABSTRACT FROM AUTHOR]

Let J ( n , m , m + 1 ) denote the incidence graph of Johnson geometry. It is well known that the Johnson graph J ( n , m ) is a halved graph of J ( n , m , m + 1 ) . Let T and T ′ be the Terwilliger algebra of J ( n , m , m + 1 ) and J ( n , m ) , respectively. In this paper, we focus on the structures of irreducible T -modules, and then completely determine T . Furthermore, we discover the relationship between irreducible modules of T and T ′ . As a result, the algebra T ′ is determined again. [ABSTRACT FROM AUTHOR]

Abstract: In Levstein and Maldonado (2007), the Terwilliger algebra of the Johnson scheme was determined when . In this paper, we determine the Terwilliger algebra for the remaining case . [Copyright &y& Elsevier]

Abstract: Folding a sequence into a multidimensional box is an important technique in multidimensional coding. In this paper, the definition of folding defined by T. Etzion is explained from an algebraic point of view, and furthermore, a necessary and sufficient condition is derived for the existence of a folding for any given shape in . [Copyright &y& Elsevier]

Abstract: In [F. Levstein, C. Maldonado, The Terwilliger algebra of the Johnson schemes, Discrete Math. 307 (2007) 1621–1635], Levstein and Maldonado computed the Terwilliger algebra of the Johnson scheme when . The distance- graph of is the Odd graph . In this paper, we determine the Terwilliger algebra of and give its basis. [Copyright &y& Elsevier]

Abstract: Let denote a -polynomial distance-regular graph with diameter and standard module . Recently, Ito and Terwilliger introduced four direct sum decompositions of ; we call these the -split decompositions of , where . In this paper we show that the ()-split decomposition and the ()-split decomposition are dual with respect to the standard Hermitian form on . We also show that the ()-split decomposition and the ()-split decomposition are dual with respect to the standard Hermitian form on . [Copyright &y& Elsevier]

Abstract: In the present paper we study the blocking number of a convex body. We determine the blocking number of the crosspolytope in . We also estimate that the blocking number of the unit ball in is at most 6, for . For a -dimensional cylinder whose base is a -dimensional convex body , we obtain a lower bound and an upper bound of its blocking number in terms of the Hadwiger covering number and the blocking number of respectively. Moreover, we introduce and study the fixing number of a convex body, we determine the exact lower and upper bounds of fixing numbers for all -dimensional convex bodies. [Copyright &y& Elsevier]

Abstract: A graph is induced matching extendable, shortly IM-extendable, if every induced matching of is included in a perfect matching of . For a nonnegative integer , a graph is called a -edge-deletable IM-extendable graph, if, for every with , is IM-extendable. In this paper, we characterize the -edge-deletable IM-extendable graphs with minimum number of edges. We show that, for a positive integer , if is a-edge-deletable IM-extendable graph on vertices, then ; furthermore, the equality holds if and only if either , or for some integer and , where is the empty graph on vertices and is the graph obtained from by replacing each vertex with a graph isomorphic to . [Copyright &y& Elsevier]

Abstract: A pair of sequences such that and is termed a WP-Bailey Pair. Upon setting in such a pair we obtain a Bailey pair. In the present paper we consider the problem of “lifting” a Bailey pair to a WP-Bailey pair, and use some of the new WP-Bailey pairs found in this way to derive some new identities between basic hypergeometric series and new single-sum and double-sum identities of the Rogers–Ramanujan–Slater type. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the existence of a super-simple (4, 5)-GDD of type and show that such a design exists if and only if , , and . [Copyright &y& Elsevier]

Abstract: In 1996, Reed proved that the domination number of every -vertex graph with minimum degree at least 3 is at most . This bound is sharp for cubic graphs if there is no restriction on connectivity. In this paper we show that for every -vertex cubic connected graph if . Note that Reed’s conjecture that for every connected cubic -vertex graph is not true. [Copyright &y& Elsevier]

Abstract: A graph is called quasi-claw-free if for any two vertices and with distance two there exists a vertex such that . This concept is a natural extension of the classical claw-free graphs. In this paper, we present two sufficient conditions for vertex pancyclicity in quasi-claw-free graphs, namely, quasilocally connected and almost locally connected graphs. Our results include some well-known results on claw-free graphs as special cases. We also give an affirmative answer to a problem proposed by Ainouche. [Copyright &y& Elsevier]

Abstract: An antichain of a well-founded quasi-order is canonical if for every ideal of , has an infinite antichain if and only if is infinite. In this paper we characterize the obstructions to having a canonical antichain. As an application we show that, under the induced subgraph relation, the class of finite graphs does not have a canonical antichain. In contrast, this class does have a canonical antichain with respect to the subgraph relation. [Copyright &y& Elsevier]

Abstract: A well known conjecture of Hadwiger asserts that is the only minor minimal graph of chromatic number greater than . In this paper, all minor minimal graphs of chromatic index greater than are determined. [Copyright &y& Elsevier]

Abstract: In this paper the existence of two types of edge-switchable -designs are investigated. It is established that the necessary conditions for a Type I switchable -design is , , and these conditions are also sufficient except for . The necessary conditions for a Type II switchable -design are , , and these conditions are also sufficient with a small number of possible exceptions. [Copyright &y& Elsevier]

Abstract: In this paper, the first five sharp upper bounds on the spectral radii of unicyclic graphs with fixed matching number are presented. The first ten spectral radii over the class of unicyclic graphs on a given number of vertices and the first four spectral radii of unicyclic graphs with perfect matchings are also given, respectively. [Copyright &y& Elsevier]

Abstract: A graph of order is called a bicyclic graph if is connected and the number of edges of is . Let be the set of all bicyclic graphs on vertices. In this paper, we obtain the first four largest Laplacian spectral radii among all the graphs in the class () together with the corresponding graphs. [Copyright &y& Elsevier]

Abstract: A graph is called chromatic-choosable if its choice number is equal to its chromatic number, namely . Ohba has conjectured that every graph satisfying is chromatic-choosable. Since each -chromatic graph is a subgraph of a complete -partite graph, we see that Ohba’s conjecture is true if and only if it is true for every complete multipartite graph. However, the only complete multipartite graphs for which Ohba’s conjecture has been verified are: , , , , and . In this paper, we show that Ohba’s conjecture is true for two new classes of complete multipartite graphs: graphs with three parts of size 3 and graphs with one part of size 4 and two parts of size 3. Namely, we prove that and (for and , respectively). [Copyright &y& Elsevier]

Abstract: Let be a simple undirected graph. Denote by (respectively, ) the number of maximal (respectively, maximum) independent sets in . Erdős and Moser raised the problem of determining the maximum value of among all graphs of order and the extremal graphs achieving this maximum value. This problem was solved by Moon and Moser. Then it was studied for many special classes of graphs, including trees, forests, bipartite graphs, connected graphs, (connected) triangle-free graphs, (connected) graphs with at most one cycle, and recently, (connected) graphs with at most cycles. In this paper we determine the second largest value of and among all graphs of order . Moreover, the extremal graphs achieving these values are also determined. [Copyright &y& Elsevier]

Abstract: For a graph , its cubicity is the minimum dimension such that is representable as the intersection graph of (axis-parallel) cubes in -dimensional space. (A -dimensional cube is a Cartesian product , where is a closed interval of the form on the real line.) Chandran et al. [L.S. Chandran, C. Mannino, G. Oriolo, On the cubicity of certain graphs, Information Processing Letters 94 (2005) 113–118] showed that for a -dimensional hypercube , . In this paper, we use the probabilistic method to show that . The parameter boxicity generalizes cubicity: the boxicity of a graph is defined as the minimum dimension such that is representable as the intersection graph of axis-parallel boxes in -dimensional space. Since for any graph , our result implies that . The problem of determining a non-trivial lower bound for is left open. [Copyright &y& Elsevier]

Abstract: We consider pattern containment and avoidance with a very tight definition that was used first in several papers more than 60 years ago. Using this definition, we prove that the monotone pattern is easier to avoid than almost any other pattern of the same length. We also show that with this definition, almost all patterns of length are avoided by the same number of permutations of length . The corresponding statements are not known to be true for more relaxed definitions of pattern containment. This is the first time we know of that expectations have been used to compare numbers of permutations avoiding certain patterns. [Copyright &y& Elsevier]

Abstract: A graph on vertices is called a Dirac graph if it has a minimum degree of at least . The distance is defined as the number of edges in a shortest path of joining and . In this paper we show that in a Dirac graph , for every small enough subset of the vertices, we can distribute the vertices of along a Hamiltonian cycle of in such a way that all but two pairs of subsequent vertices of have prescribed distances (apart from a difference of at most 1) along . More precisely we show the following. There are such that if is a Dirac graph on vertices, is an arbitrary integer with and is an arbitrary subset of the vertices of with , then for every sequence of integers with , there is a Hamiltonian cycle of and an ordering of the vertices of , , such that the vertices of are visited in this order on and we have [Copyright &y& Elsevier]

Abstract: Let denote the set of graphs of order that contain no cycles of length less than or equal to which have maximum number of edges. In this paper we consider a problem posed by several authors: does contain an cycle? We prove that the diameter of is at most , and present several results concerning the above question: the girth of is if (i) , diameter equal to and minimum degree at least 3; (ii) , and . Moreover, if we find an extremal graph of girth 8 obtained from a 3-regular complete bipartite graph subdividing its edges. (iii) We prove that if and the girth is at most . We also show that the answer to the question is negative for . [Copyright &y& Elsevier]

Abstract: For a 2-factor of a connected graph , we consider , which is the graph obtained from by removing all the edges of . If is connected, is said to be a non-separating 2-factor. In this paper we study a sufficient condition for a -regular connected graph to have such a 2-factor. As a result, we show that a -regular connected graph has a non-separating 2-factor whenever the number of vertices of does not exceed . [Copyright &y& Elsevier]

Abstract: A connected graph is -connected if there is a set of edges whose deletion leaves two components of order at least and , respectively. In this paper we present some sufficient conditions for graphs to be -connected. Furthermore, we study -connected graphs in more detail. [Copyright &y& Elsevier]

Abstract: Matching extendability is significant in graph theory and its applications. The basic notion in this direction is -extendability introduced by Plummer in 1980. Motivated by the different natures of bipartite matchings and non-bipartite matchings, this paper investigates bipartite-matching extendable (BM-extendable) graphs. A graph is said to be BM-extendable if every matching which is a perfect matching of an induced bipartite subgraph can be extended to a perfect matching. Our main results are showing that the recognition of BM-extendable graphs is co-NP-complete and characterizing some classes of BM-extendable graphs. [Copyright &y& Elsevier]

Abstract: A new homomorphism between two partially ordered sets (the III-homomorphism) and a new congruence on a poset (the III-congruence) are introduced. Some properties of these homomorphisms and congruences and their relationship to the other known homomorphisms and congruences on posets are investigated. In contrast to total algebras, there are many different ways to introduce these notions. It is usually required that the respective notions should coincide with the usual definitions whenever lattices or semilattices are treated. The present paper presents an approach which in some sense completes the hierarchy of definitions so far used. [Copyright &y& Elsevier]

Abstract: In this paper, we will survey several results on the shortest directing words of various types of nondeterministic directable automata. 1 [1] More details concerning this subject can be seen in the book: M. Ito, Algebraic Theory of Automata and Languages, World Scientific, Singapore, 2004, Chapter 8. The insertion of this sentence was suggested by one of the referees. [Copyright &y& Elsevier]

Abstract: The goal of this paper is to give a comparatively short and simple analysis of the Adyan origional group constraction (S.I. Adyan, Unsolvability of some algorithmic problems in the theory of groups, Trudy MMO 6 (1957) 231–298). [Copyright &y& Elsevier]

Abstract: An code is called equidistant code if the Hamming distance between any two codewords is d. It was proved that for any equidistant code, (, say). A necessary condition for the existence of an optimal equidistant code is that be an integer. If is not an integer, i.e. the equidistant code is not optimal, then the code with is called good equidistant code, which is obviously the best possible one among equidistant codes with parameters and q. In this paper, some constructions of good equidistant codes from balanced arrays and nested BIB designs are described. [Copyright &y& Elsevier]

Abstract: In this paper we investigate the behaviour of the gaps in numerical semigroups. We will give an explicit formula for the th gap of a semigroup generated by consecutive integers (generalizing a result due to Brauer) as well as for a special numerical semigroup of three generators. It is also proved that the number of gaps of the numerical semigroup generated by integers p and q with , in the interval is equals toWe actually give two proofs of the latter result, the first one uses the so-called Apery sets and the second one is an application of the well-known Pick''s theorem. [Copyright &y& Elsevier]

Abstract: In this paper we study recurrences concerning the combinatorial sum and the alternate sum , where , and r are integers. For example, we show that if thenWe also apply such results to investigate Bernoulli and Euler polynomials. Our approach depends heavily on an identity established by the author [A curious identity involving binomial coefficients, Integers: Electron. J. Combin. Number Theory 2 (2002), A4, 8pp]. [Copyright &y& Elsevier]