Riordan Matrix Representations of Euler’s Constant γ and Euler’s Number e

We show that the Euler-Mascheroni constant γ and Euler’s number e can both be represented as a product of a Riordan matrix and certain row and column vectors.

Groups Containing Small Locally Maximal Product-Free Sets

Let G be a group and S a nonempty subset of G. Then, S is product-free if ab∉S for all a,b∈S. We say S is a locally maximal product-free set if S is product-free and not properly contained in any other product-free set. It is natural to ask whether it is possible to determine the smallest possible size of a locally maximal product-free set in G. Alternatively, given a positive integer k, one can ask the following: what is the largest integer nk such that there is a group of order nk with a locally maximal product-free set of size k? The groups containing locally maximal product-free sets of sizes 1 and 2 are known, and it has been conjectured that n3=24. The purpose of this paper is to prove this conjecture and hence show that the list of known locally maximal product-free sets of size 3 is complete. We also report some experimental observations about the sequence nk.

A Generalized Inverse Binomial Summation Theorem and Some Hypergeometric Transformation Formulas

A generalized binomial theorem is developed in terms of Bell polynomials and by applying this identity some sums involving inverse binomial coefficient are calculated. A technique is derived for calculating a class of hypergeometric transformation formulas and also some curious q series identities.

Combinatorial Analysis of a Subtraction Game on Graphs

We define a two-player combinatorial game in which players take alternate turns; each turn consists of deleting a vertex of a graph, together with all the edges containing such vertex. If any vertex became isolated by a player’s move then it would also be deleted. A player wins the game when the other player has no moves available. We study this game under various viewpoints: by finding specific strategies for certain families of graphs, through using properties of a graph’s automorphism group, by writing a program to look at Sprague-Grundy numbers, and by studying the game when played on random graphs. When analyzing Grim played on paths, using the Sprague-Grundy function, we find a connection to a standing open question about Octal games.

On Self-Centeredness of Product of Graphs

A graph G is said to be a self-centered graph if the eccentricity of every vertex of the graph is the same. In other words, a graph is a self-centered graph if radius and diameter of the graph are equal. In this paper, self-centeredness of strong product, co-normal product, and lexicographic product of graphs is studied in detail. The necessary and sufficient conditions for these products of graphs to be a self-centered graph are also discussed. The distance between any two vertices in the co-normal product of a finite number of graphs is also computed analytically.

A New Relationship between Block Designs

We propose a procedure of constructing new block designs starting from a given one by looking at the intersections of its blocks with various sets and grouping those sets according to the structure of the intersections. We introduce a symmetric relationship of friendship between block designs built on a set V and consider families of block designs where all designs are friends of each other, the so-called friendly families. We show that a friendly family admits a partial ordering. Furthermore, we exhibit a map from the power set of V, partially ordered by inclusion, to a friendly family of a particular type which preserves the partial order.

Some Nonexistence and Asymptotic Existence Results for Weighing Matrices

Orthogonal designs and weighing matrices have many applications in areas such as coding theory, cryptography, wireless networking, and communication. In this paper, we first show that if positive integer k cannot be written as the sum of three integer squares, then there does not exist any skew-symmetric weighing matrix of order 4n and weight k, where n is an odd positive integer. Then we show that, for any square k, there is an integer N(k) such that, for each n≥N(k), there is a symmetric weighing matrix of order n and weight k. Moreover, we improve some of the asymptotic existence results for weighing matrices obtained by Eades, Geramita, and Seberry.

Starter Labelling of k-Windmill Graphs with Small Defects

A graph on 2n vertices can be starter-labelled, if the vertices can be given labels from the nonzero elements of the additive group Z2n+1 such that each label i, either i or i-1, is assigned to exactly two vertices and the two vertices are separated by either i edges or i-1 edges, respectively. Mendelsohn and Shalaby have introduced Skolem-labelled graphs and determined the conditions of k-windmills to be Skolem-labelled. In this paper, we introduce starter-labelled graphs and obtain necessary and sufficient conditions for starter and minimum hooked starter labelling of all k-windmills.

A Formula for the Reliability of a d-Dimensional Consecutive-k-out-of-n:F System

We derive a formula for the reliability of a d-dimensional consecutive-k-out-of-n:F system, that is, a formula for the probability that an n1×⋯×nd array whose entries are (independently of each other) 0 with probability p and 1 with probability q=1-p does not include a contiguous s1×⋯×sd subarray whose every entry is 1.

On 3-Regular Bipancyclic Subgraphs of Hypercubes

The n-dimensional hypercube Qn is bipancyclic; that is, it contains a cycle of every even length from 4 to 2n. In this paper, we prove that Qn (n≥3) contains a 3-regular, 3-connected, bipancyclic subgraph with l vertices for every even l from 8 to 2n except 10.