International Journal of Combinatorics The latest articles from Hindawi © 2017 , Hindawi Limited . All rights reserved. Riordan Matrix Representations of Euler’s Constant and Euler’s Number Sun, 06 Nov 2016 14:22:43 +0000 We show that the Euler-Mascheroni constant and Euler’s number can both be represented as a product of a Riordan matrix and certain row and column vectors. Edray Herber Goins and Asamoah Nkwanta Copyright © 2016 Edray Herber Goins and Asamoah Nkwanta. All rights reserved. Groups Containing Small Locally Maximal Product-Free Sets Thu, 13 Oct 2016 11:22:14 +0000 Let be a group and a nonempty subset of . Then, is product-free if for all . We say is a locally maximal product-free set if 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 . Alternatively, given a positive integer , one can ask the following: what is the largest integer such that there is a group of order with a locally maximal product-free set of size ? The groups containing locally maximal product-free sets of sizes and are known, and it has been conjectured that . 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 . Chimere S. Anabanti and Sarah B. Hart Copyright © 2016 Chimere S. Anabanti and Sarah B. Hart. All rights reserved. A Generalized Inverse Binomial Summation Theorem and Some Hypergeometric Transformation Formulas Tue, 27 Sep 2016 09:22:15 +0000 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 series identities. S. M. Ripon Copyright © 2016 S. M. Ripon. All rights reserved. Combinatorial Analysis of a Subtraction Game on Graphs Mon, 29 Aug 2016 07:10:54 +0000 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. Richard Adams, Janae Dixon, Jennifer Elder, Jamie Peabody, Oscar Vega, and Karen Willis Copyright © 2016 Richard Adams et al. All rights reserved. On Self-Centeredness of Product of Graphs Sun, 07 Aug 2016 07:51:09 +0000 A graph 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. Priyanka Singh and Pratima Panigrahi Copyright © 2016 Priyanka Singh and Pratima Panigrahi. All rights reserved. A New Relationship between Block Designs Thu, 28 Jul 2016 06:22:40 +0000 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 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 , partially ordered by inclusion, to a friendly family of a particular type which preserves the partial order. Alexander Shramchenko and Vasilisa Shramchenko Copyright © 2016 Alexander Shramchenko and Vasilisa Shramchenko. All rights reserved. Some Nonexistence and Asymptotic Existence Results for Weighing Matrices Sun, 13 Mar 2016 08:33:50 +0000 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 cannot be written as the sum of three integer squares, then there does not exist any skew-symmetric weighing matrix of order and weight , where is an odd positive integer. Then we show that, for any square , there is an integer such that, for each , there is a symmetric weighing matrix of order and weight . Moreover, we improve some of the asymptotic existence results for weighing matrices obtained by Eades, Geramita, and Seberry. Ebrahim Ghaderpour Copyright © 2016 Ebrahim Ghaderpour. All rights reserved. Starter Labelling of -Windmill Graphs with Small Defects Mon, 17 Aug 2015 07:40:13 +0000 A graph on vertices can be starter-labelled, if the vertices can be given labels from the nonzero elements of the additive group such that each label , either or , is assigned to exactly two vertices and the two vertices are separated by either edges or edges, respectively. Mendelsohn and Shalaby have introduced Skolem-labelled graphs and determined the conditions of -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 -windmills. Farej Omer and Nabil Shalaby Copyright © 2015 Farej Omer and Nabil Shalaby. All rights reserved. A Formula for the Reliability of a -Dimensional Consecutive--out-of-:F System Mon, 13 Jul 2015 10:58:05 +0000 We derive a formula for the reliability of a -dimensional consecutive--out-of-:F system, that is, a formula for the probability that an array whose entries are (independently of each other) 0 with probability and 1 with probability does not include a contiguous subarray whose every entry is 1. Simon Cowell Copyright © 2015 Simon Cowell. All rights reserved. On 3-Regular Bipancyclic Subgraphs of Hypercubes Tue, 05 May 2015 09:36:04 +0000 The -dimensional hypercube is bipancyclic; that is, it contains a cycle of every even length from 4 to . In this paper, we prove that contains a 3-regular, 3-connected, bipancyclic subgraph with vertices for every even from 8 to except 10. Y. M. Borse and S. R. Shaikh Copyright © 2015 Y. M. Borse and S. R. Shaikh. All rights reserved. On Evenly-Equitable, Balanced Edge-Colorings and Related Notions Wed, 04 Mar 2015 06:53:37 +0000 A graph is said to be even if all vertices of have even degree. Given a -edge-coloring of a graph , for each color let denote the spanning subgraph of in which the edge-set contains precisely the edges colored . A -edge-coloring of is said to be an -edge-coloring if for each color , is an even graph. A -edge-coloring of is said to be evenly-equitable if for each color , is an even graph, and for each vertex and for any pair of colors , . For any pair of vertices let be the number of edges between and in (we allow , where denotes a loop incident with ). A -edge-coloring of is said to be balanced if for all pairs of colors and and all pairs of vertices and (possibly ), . Hilton proved that each even graph has an evenly-equitable -edge-coloring for each . In this paper we extend this result by finding a characterization for graphs that have an evenly-equitable, balanced -edge-coloring for each . Correspondingly we find a characterization for even graphs to have an evenly-equitable, balanced 2-edge-coloring. Then we give an instance of how evenly-equitable, balanced edge-colorings can be used to determine if a certain fairness property of factorizations of some regular graphs is satisfied. Finally we indicate how different fairness notions on edge-colorings interact with each other. Aras Erzurumluoğlu and C. A. Rodger Copyright © 2015 Aras Erzurumluoğlu and C. A. Rodger. All rights reserved. The Class of -Cliqued Graphs: Eigen-Bi-Balanced Characteristic, Designs, and an Entomological Experiment Mon, 02 Mar 2015 09:52:57 +0000 Much research has involved the consideration of graphs which have subgraphs of a particular kind, such as cliques. Known classes of graphs which are eigen-bi-balanced, that is, they have a pair a, b of nonzero distinct eigenvalues, whose sum and product are integral, have been investigated. In this paper we will define a new class of graphs, called q-cliqued graphs, on vertices, which contain cliques each of order connected to a central vertex, and then prove that these -cliqued graphs are eigen-bi-balanced with respect to a conjugate pair whose sum is and product . These graphs can be regarded as design graphs, and we use a specific example in an entomological experiment. Paul August Winter, Carol Lynne Jessop, and Costas Zachariades Copyright © 2015 Paul August Winter et al. All rights reserved. Hamilton Paths and Cycles in Varietal Hypercube Networks with Mixed Faults Thu, 22 Jan 2015 13:47:02 +0000 This paper considers the varietal hypercube network with mixed faults and shows that contains a fault-free Hamilton cycle provided faults do not exceed for and contains a fault-free Hamilton path between any pair of vertices provided faults do not exceed for . The proof is based on an inductive construction. Jian-Guang Zhou and Jun-Ming Xu Copyright © 2015 Jian-Guang Zhou and Jun-Ming Xu. All rights reserved. Maximal Midpoint-Free Subsets of Integers Tue, 20 Jan 2015 12:58:00 +0000 A set is midpoint-free if no ordered triple satisfies and . Midpoint-free subsets of and are studied, with emphasis on those sets characterized by restrictions on the base digits of their elements when , and with particular attention to maximal midpoint-free subsets with . Roger B. Eggleton Copyright © 2015 Roger B. Eggleton. All rights reserved. Betweenness Centrality in Some Classes of Graphs Thu, 25 Dec 2014 13:26:55 +0000 There are several centrality measures that have been introduced and studied for real-world networks. They account for the different vertex characteristics that permit them to be ranked in order of importance in the network. Betweenness centrality is a measure of the influence of a vertex over the flow of information between every pair of vertices under the assumption that information primarily flows over the shortest paths between them. In this paper we present betweenness centrality of some important classes of graphs. Sunil Kumar Raghavan Unnithan, Balakrishnan Kannan, and Madambi Jathavedan Copyright © 2014 Sunil Kumar Raghavan Unnithan et al. All rights reserved. On Some Bounds and Exact Formulae for Connective Eccentric Indices of Graphs under Some Graph Operations Wed, 24 Dec 2014 11:22:31 +0000 The connective eccentric index of a graph is a topological index involving degrees and eccentricities of vertices of the graph. In this paper, we have studied the connective eccentric index for double graph and double cover. Also we give the connective eccentric index for some graph operations such as joins, symmetric difference, disjunction, and splice of graphs. Nilanjan De, Anita Pal, and Sk. Md. Abu Nayeem Copyright © 2014 Nilanjan De et al. All rights reserved. Extremal Unimodular Lattices in Dimension 36 Sun, 30 Nov 2014 16:47:35 +0000 New extremal odd unimodular lattices in dimension 36 are constructed. Some new odd unimodular lattices in dimension 36 with long shadows are also constructed. Masaaki Harada Copyright © 2014 Masaaki Harada. All rights reserved. On the General Erdős-Turán Conjecture Mon, 17 Nov 2014 09:34:43 +0000 The general Erdős-Turán conjecture states that if is an infinite, strictly increasing sequence of natural numbers whose general term satisfies , for some constant and for all , then the number of representations functions of is unbounded. Here, we introduce the function , giving the minimum of the maximal number of representations of a finite sequence of natural numbers satisfying for all . We show that is an increasing function of and that the general Erdős-Turán conjecture is equivalent to . We also compute some values of . We further introduce and study the notion of capacity, which is related to the function by the fact that is the capacity of the set of squares of positive integers, but which is also of intrinsic interest. Georges Grekos, Labib Haddad, Charles Helou, and Jukka Pihko Copyright © 2014 Georges Grekos et al. All rights reserved. Characterizing Finite Groups Using the Sum of the Orders of the Elements Thu, 13 Nov 2014 09:32:41 +0000 We give characterizations of various infinite sets of finite groups under the assumption that and the subgroups of satisfy certain properties involving the sum of the orders of the elements of and . Additionally, we investigate the possible values for the sum of the orders of the elements of . Joshua Harrington, Lenny Jones, and Alicia Lamarche Copyright © 2014 Joshua Harrington et al. All rights reserved. Domination Polynomials of k-Tree Related Graphs Tue, 11 Nov 2014 07:14:50 +0000 Let G be a simple graph of order n. The domination polynomial of G is the polynomial , where d(G, i) is the number of dominating sets of G of size i and γ(G) is the domination number of G. In this paper, we study the domination polynomials of several classes of k-tree related graphs. Also, we present families of these kinds of graphs, whose domination polynomials have no nonzero real roots. Somayeh Jahari and Saeid Alikhani Copyright © 2014 Somayeh Jahari and Saeid Alikhani. All rights reserved. Necklaces, Self-Reciprocal Polynomials, and -Cycles Sun, 09 Nov 2014 07:04:34 +0000 Let be a positive integer and a prime power. Consider necklaces consisting of beads, each of which has one of the given colors. A primitive -orbit is an equivalence class of necklaces closed under rotation. A -orbit is self-complementary when it is closed under an assigned color matching. In the work of Miller (1978), it is shown that there is a 1-1 correspondence between the set of primitive, self-complementary -orbits and that of self-reciprocal irreducible monic (srim) polynomials of degree . Let be a positive integer relatively prime to . A -cycle mod is a finite sequence of nonnegative integers closed under multiplication by . In the work of Wan (2003), it is shown that -cycles mod are closely related to monic irreducible divisors of . Here, we show that: (1) -cycles can be used to obtain information about srim polynomials; (2) there are correspondences among certain -cycles and -orbits; (3) there are alternative proofs of Miller's results in the work of Miller (1978) based on the use of -cycles. Umarin Pintoptang, Vichian Laohakosol, and Suton Tadee Copyright © 2014 Umarin Pintoptang et al. All rights reserved. Some Properties of the Intersection Graph for Finite Commutative Principal Ideal Rings Thu, 25 Sep 2014 07:44:53 +0000 Let R be a commutative finite principal ideal ring with unity, and let G(R) be the simple graph consisting of nontrivial proper ideals of R as vertices such that two vertices I and J are adjacent if they have nonzero intersection. In this paper we continue the work done by Abu Osba. We calculate the radius, eccentricity, domination number, independence number, geodetic number, and the hull number for this graph. We also determine when G(R) is chordal. Finally, we study some properties of the complement graph of G(R). Emad Abu Osba, Salah Al-Addasi, and Omar Abughneim Copyright © 2014 Emad Abu Osba et al. All rights reserved. Integral Eigen-Pair Balanced Classes of Graphs with Their Ratio, Asymptote, Area, and Involution-Complementary Aspects Tue, 23 Sep 2014 09:08:46 +0000 The association of integers, conjugate pairs, and robustness with the eigenvalues of graphs provides the motivation for the following definitions. A class of graphs, with the property that, for each graph (member) of the class, there exists a pair of nonzero, distinct eigenvalues, whose sum and product are integral, is said to be eigen-bibalanced. If the ratio is a function , of the order of the graphs in this class, then we investigate its asymptotic properties. Attaching the average degree to the Riemann integral of this ratio allowed for the evaluation of eigen-balanced areas of classes of graphs. Complete graphs on vertices are eigen-bibalanced with the eigen-balanced ratio which is asymptotic to the constant value of −1. Its eigen-balanced area is —we show that this is the maximum area for most known classes of eigen-bibalanced graphs. We also investigate the class of eigen-bibalanced graphs, whose class of complements gives rise to an eigen-balanced asymptote that is an involution and the effect of the asymptotic ratio on the energy of the graph theoretical representation of molecules. Paul August Winter and Carol Lynne Jessop Copyright © 2014 Paul August Winter and Carol Lynne Jessop. All rights reserved. Modular Leech Trees of Order at Most 8 Thu, 18 Sep 2014 10:54:22 +0000 In 1975, John Leech asked when can the edges of a tree on vertices be labeled with positive integers such that the sums along the paths are exactly the integers . He found five such trees, and no additional trees have been discovered since. In 2011 Leach and Walsh introduced the idea of labeling trees with elements of the group where and examined the cases for . In this paper we show that no modular Leech trees of order 7 exist, and we find all modular Leech trees of order 8. David Leach Copyright © 2014 David Leach. All rights reserved. Midpoint-Free Subsets of the Real Numbers Tue, 26 Aug 2014 08:07:57 +0000 A set of reals is midpoint-free if it has no subset such that and . If and is midpoint-free, it is a maximal midpoint-free subset of if there is no midpoint-free set such that . In each of the cases , we determine two maximal midpoint-free subsets of characterised by digit constraints on the base 3 representations of their members. Roger B. Eggleton Copyright © 2014 Roger B. Eggleton. All rights reserved. Normal Edge-Transitive Cayley Graphs of the Group Tue, 19 Aug 2014 00:00:00 +0000 A Cayley graph of a group is called normal edge-transitive if the normalizer of the right representation of the group in the automorphism of the Cayley graph acts transitively on the set of edges of the graph. In this paper, we determine all connected normal edge-transitive Cayley graphs of the group . A. Assari and F. Sheikhmiri Copyright © 2014 A. Assari and F. Sheikhmiri. All rights reserved. On the Genus of the Zero-Divisor Graph of Tue, 22 Jul 2014 08:04:37 +0000 Let be a commutative ring with identity. The zero-divisor graph of , denoted , is the simple graph whose vertices are the nonzero zero-divisors of , and two distinct vertices and are linked by an edge if and only if . The genus of a simple graph is the smallest integer such that can be embedded into an orientable surface . In this paper, we determine that the genus of the zero-divisor graph of , the ring of integers modulo , is two or three. Huadong Su and Pailing Li Copyright © 2014 Huadong Su and Pailing Li. All rights reserved. A Weighted Regularity Lemma with Applications Thu, 19 Jun 2014 11:48:40 +0000 We prove an extension of the regularity lemma with vertex and edge weights which in principle can be applied for arbitrary graphs. The applications involve random graphs and a weighted version of the Erdős-Stone theorem. We also provide means to handle the otherwise uncontrolled exceptional set. Béla Csaba and András Pluhár Copyright © 2014 Béla Csaba and András Pluhár. All rights reserved. Decomposition Formulas for Triple -Hypergeometric Functions Thu, 15 May 2014 12:45:07 +0000 In the spirit of Hasanov, Srivastava, and Turaev (2006), we introduce new inverse operators together with a more general operator and find a summation formula for the last one. Based on these operators and the earlier known -analogues of the Burchnall-Chaundy operators, we find 15 symbolic operator formulas. Then, 10 expansions for the -analogues of Srivastava’s three triple hypergeometric functions in terms of -hypergeometric and -Kampé de Fériet functions are derived. These expansions readily reduce to 10 new expansions for the three triple Srivastava hypergeometric functions in terms of hypergeometric and Kampé de Fériet functions. Thomas Ernst Copyright © 2014 Thomas Ernst. All rights reserved. The Terminal Hosoya Polynomial of Some Families of Composite Graphs Wed, 16 Apr 2014 08:34:46 +0000 Let be a connected graph and let be the set of pendent vertices of . The terminal Hosoya polynomial of is defined as , where denotes the distance between the pendent vertices and . In this note paper we obtain closed formulae for the terminal Hosoya polynomial of rooted product graphs and corona product graphs. Emeric Deutsch and Juan Alberto Rodríguez-Velázquez Copyright © 2014 Emeric Deutsch and Juan Alberto Rodríguez-Velázquez. All rights reserved.