International Journal of Combinatorics
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Open Access journal
ISSN (Print) 1687-9163 - ISSN (Online) 1687-9171
Published by Hindawi [334 journals]
- Riordan Matrix Representations of Euler’s Constant and Euler’s
Abstract: 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.
PubDate: Sun, 06 Nov 2016 14:22:43 +000
- Groups Containing Small Locally Maximal Product-Free Sets
Abstract: 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 .
PubDate: Thu, 13 Oct 2016 11:22:14 +000
- A Generalized Inverse Binomial Summation Theorem and Some Hypergeometric
Abstract: 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.
PubDate: Tue, 27 Sep 2016 09:22:15 +000
- Combinatorial Analysis of a Subtraction Game on Graphs
Abstract: 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.
PubDate: Mon, 29 Aug 2016 07:10:54 +000
- On Self-Centeredness of Product of Graphs
Abstract: 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.
PubDate: Sun, 07 Aug 2016 07:51:09 +000
- A New Relationship between Block Designs
Abstract: 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.
PubDate: Thu, 28 Jul 2016 06:22:40 +000
- Some Nonexistence and Asymptotic Existence Results for Weighing Matrices
Abstract: 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.
PubDate: Sun, 13 Mar 2016 08:33:50 +000
- Starter Labelling of -Windmill Graphs with Small Defects
Abstract: 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.
PubDate: Mon, 17 Aug 2015 07:40:13 +000
- A Formula for the Reliability of a -Dimensional Consecutive--out-of-:F
Abstract: 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.
PubDate: Mon, 13 Jul 2015 10:58:05 +000
- On 3-Regular Bipancyclic Subgraphs of Hypercubes
Abstract: 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.
PubDate: Tue, 05 May 2015 09:36:04 +000
- On Evenly-Equitable, Balanced Edge-Colorings and Related Notions
Abstract: 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.
PubDate: Wed, 04 Mar 2015 06:53:37 +000
- The Class of -Cliqued Graphs: Eigen-Bi-Balanced Characteristic, Designs,
and an Entomological Experiment
Abstract: 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.
PubDate: Mon, 02 Mar 2015 09:52:57 +000
- Hamilton Paths and Cycles in Varietal Hypercube Networks with Mixed Faults
Abstract: 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.
PubDate: Thu, 22 Jan 2015 13:47:02 +000
- Maximal Midpoint-Free Subsets of Integers
Abstract: 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 .
PubDate: Tue, 20 Jan 2015 12:58:00 +000
- Betweenness Centrality in Some Classes of Graphs
Abstract: 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.
PubDate: Thu, 25 Dec 2014 13:26:55 +000
- On Some Bounds and Exact Formulae for Connective Eccentric Indices of
Graphs under Some Graph Operations
Abstract: 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.
PubDate: Wed, 24 Dec 2014 11:22:31 +000
- Extremal Unimodular Lattices in Dimension 36
Abstract: New extremal odd unimodular lattices in dimension 36 are constructed. Some new odd unimodular lattices in dimension 36 with long shadows are also constructed.
PubDate: Sun, 30 Nov 2014 16:47:35 +000
- On the General Erdős-Turán Conjecture
Abstract: 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.
PubDate: Mon, 17 Nov 2014 09:34:43 +000
- Characterizing Finite Groups Using the Sum of the Orders of the Elements
Abstract: 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 .
PubDate: Thu, 13 Nov 2014 09:32:41 +000
- Domination Polynomials of k-Tree Related Graphs
Abstract: 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.
PubDate: Tue, 11 Nov 2014 07:14:50 +000
- Necklaces, Self-Reciprocal Polynomials, and -Cycles
Abstract: 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.
PubDate: Sun, 09 Nov 2014 07:04:34 +000
- Some Properties of the Intersection Graph for Finite Commutative Principal
Abstract: 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).
PubDate: Thu, 25 Sep 2014 07:44:53 +000
- Integral Eigen-Pair Balanced Classes of Graphs with Their Ratio,
Asymptote, Area, and Involution-Complementary Aspects
Abstract: 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.
PubDate: Tue, 23 Sep 2014 09:08:46 +000
- Modular Leech Trees of Order at Most 8
Abstract: 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.
PubDate: Thu, 18 Sep 2014 10:54:22 +000
- Midpoint-Free Subsets of the Real Numbers
Abstract: 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.
PubDate: Tue, 26 Aug 2014 08:07:57 +000
- Normal Edge-Transitive Cayley Graphs of the Group
Abstract: 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 .
PubDate: Tue, 19 Aug 2014 00:00:00 +000
- On the Genus of the Zero-Divisor Graph of
Abstract: 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.
PubDate: Tue, 22 Jul 2014 08:04:37 +000
- A Weighted Regularity Lemma with Applications
Abstract: 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.
PubDate: Thu, 19 Jun 2014 11:48:40 +000
- Decomposition Formulas for Triple -Hypergeometric Functions
Abstract: 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.
PubDate: Thu, 15 May 2014 12:45:07 +000
- The Terminal Hosoya Polynomial of Some Families of Composite Graphs
Abstract: 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.
PubDate: Wed, 16 Apr 2014 08:34:46 +000