Journal of Discrete Mathematics http://www.hindawi.com The latest articles from Hindawi Publishing Corporation © 2014 , Hindawi Publishing Corporation . All rights reserved. Radio Numbers of Certain -Distant Trees Mon, 15 Dec 2014 08:53:55 +0000 http://www.hindawi.com/journals/jdm/2014/486354/ Radio coloring of a graph with diameter is an assignment of positive integers to the vertices of such that , where and are any two distinct vertices of and is the distance between and . The number max is called the span of . The minimum of spans over all radio colorings of is called radio number of , denoted by . An m-distant tree T is a tree in which there is a path of maximum length such that every vertex in is at the most distance from . This path is called a central path. For every tree , there is an integer such that is a -distant tree. In this paper, we determine the radio number of some -distant trees for any positive integer , and as a consequence of it, we find the radio number of a class of 1-distant trees (or caterpillars). Srinivasa Rao Kola and Pratima Panigrahi Copyright © 2014 Srinivasa Rao Kola and Pratima Panigrahi. All rights reserved. Knight’s Tours on Rectangular Chessboards Using External Squares Tue, 09 Dec 2014 00:10:09 +0000 http://www.hindawi.com/journals/jdm/2014/210892/ The classic puzzle of finding a closed knight’s tour on a chessboard consists of moving a knight from square to square in such a way that it lands on every square once and returns to its starting point. The 8 × 8 chessboard can easily be extended to rectangular boards, and in 1991, A. Schwenk characterized all rectangular boards that have a closed knight’s tour. More recently, Demaio and Hippchen investigated the impossible boards and determined the fewest number of squares that must be removed from a rectangular board so that the remaining board has a closed knight’s tour. In this paper we define an extended closed knight’s tour for a rectangular chessboard as a closed knight’s tour that includes all squares of the board and possibly additional squares beyond the boundaries of the board and answer the following question: how many squares must be added to a rectangular chessboard so that the new board has a closed knight’s tour? Grady Bullington, Linda Eroh, Steven J. Winters, and Garry L. Johns Copyright © 2014 Grady Bullington et al. All rights reserved. Distance Degree Regular Graphs and Distance Degree Injective Graphs: An Overview Mon, 08 Dec 2014 08:34:52 +0000 http://www.hindawi.com/journals/jdm/2014/358792/ The distance from a vertex of to a vertex is the length of shortest to path. The eccentricity of is the distance to a farthest vertex from . If , we say that is an eccentric vertex of . The radius is the minimum eccentricity of the vertices, whereas the diameter is the maximum eccentricity. A vertex is a central vertex if , and a vertex is a peripheral vertex if . A graph is self-centered if every vertex has the same eccentricity; that is, . The distance degree sequence (dds) of a vertex in a graph is a list of the number of vertices at distance in that order, where denotes the eccentricity of in . Thus, the sequence is the distance degree sequence of the vertex in where denotes the number of vertices at distance from . The concept of distance degree regular (DDR) graphs was introduced by Bloom et al., as the graphs for which all vertices have the same distance degree sequence. By definition, a DDR graph must be a regular graph, but a regular graph may not be DDR. A graph is distance degree injective (DDI) graph if no two vertices have the same distance degree sequence. DDI graphs are highly irregular, in comparison with the DDR graphs. In this paper we present an exhaustive review of the two concepts of DDR and DDI graphs. The paper starts with an insight into all distance related sequences and their applications. All the related open problems are listed. Medha Itagi Huilgol Copyright © 2014 Medha Itagi Huilgol. All rights reserved. Eccentric Connectivity and Zagreb Coindices of the Generalized Hierarchical Product of Graphs Thu, 27 Nov 2014 13:49:13 +0000 http://www.hindawi.com/journals/jdm/2014/292679/ Formulas for calculations of the eccentric connectivity index and Zagreb coindices of graphs under generalized hierarchical product are presented. As an application, explicit formulas for eccentric connectivity index and Zagreb coindices of some chemical graphs are obtained. M. Tavakoli, F. Rahbarnia, and A. R. Ashrafi Copyright © 2014 M. Tavakoli et al. All rights reserved. New Classes of Graceful Trees Sun, 23 Nov 2014 12:30:27 +0000 http://www.hindawi.com/journals/jdm/2014/194759/ Graceful labeling is one of the most researched problems. One of the earliest results is that caterpillars are graceful. We show that caterpillars connected to a vertex recursively satisfying certain conditions are also graceful. Md. Forhad Hossain, Md. Momin Al Aziz, and M. Kaykobad Copyright © 2014 Md. Forhad Hossain et al. All rights reserved. On the Barycentric Labeling of Certain Graphs Sun, 16 Nov 2014 12:44:52 +0000 http://www.hindawi.com/journals/jdm/2014/482635/ Let be an abelian group. A graph is called -magic if there exists edge labeling such that the induced vertex set labeling , defined by , where the sum is over all edges in , is a constant map. A graph is -barycentric-magic (or has -barycentric labeling) if is -magic and also satisfies for all and for some vertex adjacent to . In this paper we consider some graphs and characterize all for which is -barycentric-magic. Saeid Alikhani and Zeynab Amirzadeh Copyright © 2014 Saeid Alikhani and Zeynab Amirzadeh. All rights reserved. Hermitian Self-Orthogonal Constacyclic Codes over Finite Fields Wed, 12 Nov 2014 07:07:21 +0000 http://www.hindawi.com/journals/jdm/2014/985387/ Necessary and sufficient conditions for the existence of Hermitian self-orthogonal constacyclic codes of length over a finite field , coprime to , are found. The defining sets and corresponding generator polynomials of these codes are also characterised. A formula for the number of Hermitian self-orthogonal constacyclic codes of length over a finite field is obtained. Conditions for the existence of numerous MDS Hermitian self-orthogonal constacyclic codes are obtained. The defining set and the number of such MDS codes are also found. Amita Sahni and Poonam Trama Sehgal Copyright © 2014 Amita Sahni and Poonam Trama Sehgal. All rights reserved. Smarandache Curves according to Curves on a Spacelike Surface in Minkowski 3-Space Thu, 16 Oct 2014 09:20:29 +0000 http://www.hindawi.com/journals/jdm/2014/829581/ We introduce Smarandache curves according to the Lorentzian Darboux frame of a curve on spacelike surface in Minkowski 3-space . Also, we obtain the Sabban frame and the geodesic curvature of the Smarandache curves and give some characterizations on the curves when the curve α is an asymptotic curve or a principal curve. And we give an example to illustrate these curves. Ufuk Ozturk and Esra Betul Koc Ozturk Copyright © 2014 Ufuk Ozturk and Esra Betul Koc Ozturk. All rights reserved. Counting Irreducible Polynomials of Degree over and Generating Goppa Codes Using the Lattice of Subfields of Thu, 18 Sep 2014 00:00:00 +0000 http://www.hindawi.com/journals/jdm/2014/263179/ The problem of finding the number of irreducible monic polynomials of degree over is considered in this paper. By considering the fact that an irreducible polynomial of degree over has a root in a subfield of if and only if , we show that Gauss’s formula for the number of monic irreducible polynomials can be derived by merely considering the lattice of subfields of . We also use the lattice of subfields of to determine if it is possible to generate a Goppa code using an element lying in a proper subfield of . Kondwani Magamba and John A. Ryan Copyright © 2014 Kondwani Magamba and John A. Ryan. All rights reserved. Vague Filters of Residuated Lattices Wed, 10 Sep 2014 05:28:00 +0000 http://www.hindawi.com/journals/jdm/2014/120342/ Notions of vague filters, subpositive implicative vague filters, and Boolean vague filters of a residuated lattice are introduced and some related properties are investigated. The characterizations of (subpositive implicative, Boolean) vague filters is obtained. We prove that the set of all vague filters of a residuated lattice forms a complete lattice and we find its distributive sublattices. The relation among subpositive implicative vague filters and Boolean vague filters are obtained and it is proved that subpositive implicative vague filters are equivalent to Boolean vague filters. Shokoofeh Ghorbani Copyright © 2014 Shokoofeh Ghorbani. All rights reserved. Modularity in the Semilattice of ω-Words Thu, 15 May 2014 11:29:53 +0000 http://www.hindawi.com/journals/jdm/2014/891760/ A partial ordering of ω-words can be introduced with regard to whether an ω-word can be transformed into another by a Mealy machine. It is known that the poset of ω-words that is introduced by this ordering is a join-semilattice. The width of this join-semilattice has the power of continuum while the depth is at least . We have created a technique for proving that power-characteristic ω-words are incomparable. We use this technique to show that this join-semilattice is not modular. Jānis Buls and Edmunds Cers Copyright © 2014 Jānis Buls and Edmunds Cers. All rights reserved. -Approximation: A New Approach to Algebraic Approximation Mon, 28 Apr 2014 14:04:33 +0000 http://www.hindawi.com/journals/jdm/2014/909684/ We intend to study a new class of algebraic approximations, called -approximations, and their properties. We have shown that -approximations can be used for applied problems which cannot be modeled by inclusion based approximations. Also, in this work, we studied a subclass of -approximations, called -approximations, and showed that this subclass preserves most of the properties of inclusion based approximations but is not necessarily inclusionbased. The paper concludes by studying some basic operations on -approximations and counting the number of -min functions. M. R. Hooshmandasl, A. Shakiba, A. K. Goharshady, and A. Karimi Copyright © 2014 M. R. Hooshmandasl et al. All rights reserved. Chaotification for Partial Difference Equations via Controllers Thu, 13 Mar 2014 12:25:46 +0000 http://www.hindawi.com/journals/jdm/2014/538423/ Chaotification problems of partial difference equations are studied. Two chaotification schemes are established by utilizing the snap-back repeller theory of general discrete dynamical systems, and all the systems are proved to be chaotic in the sense of both Li-Yorke and Devaney. An example is provided to illustrate the theoretical results with computer simulations. Wei Liang, Yuming Shi, and Zongcheng Li Copyright © 2014 Wei Liang et al. All rights reserved. On Some Numbers Related to Extremal Combinatorial Sum Problems Mon, 03 Mar 2014 15:53:13 +0000 http://www.hindawi.com/journals/jdm/2014/979171/ Let n, d, and r be three integers such that . Chiaselotti (2002) defined as the minimum number of the nonnegative partial sums with d summands of a sum , where are n real numbers arbitrarily chosen in such a way that r of them are nonnegative and the remaining are negative. Chiaselotti (2002) and Chiaselotti et al. (2008) determine the values of for particular infinite ranges of the integer parameters n, d, and r. In this paper we continue their approach on this problem and we prove the following results: (i) for all values of n, d, and r such that ; (ii) D. Petrassi Copyright © 2014 D. Petrassi. All rights reserved. The Concept of -Cycle and Applications Wed, 26 Feb 2014 12:54:51 +0000 http://www.hindawi.com/journals/jdm/2014/823567/ The concept of -cycle is investigated for its properties and applications. Connections with irreducible polynomials over a finite field are established with emphases on the notions of order and degree. The results are applied to deduce new results about primitive and self-reciprocal polynomials. Umarin Pintoptang, Suton Tadee, and Vichian Laohakosol Copyright © 2014 Umarin Pintoptang et al. All rights reserved. A Characterization of 2-Tree Proper Interval 3-Graphs Sun, 23 Feb 2014 07:37:03 +0000 http://www.hindawi.com/journals/jdm/2014/143809/ An interval p-graph is the intersection graph of a collection of intervals which have been colored with p different colors with edges corresponding to nonempty intersection of intervals from different color classes. We characterize the class of 2-trees which are interval 3-graphs via a list of three graphs and three infinite families of forbidden induced subgraphs. David E. Brown and Breeann M. Flesch Copyright © 2014 David E. Brown and Breeann M. Flesch. All rights reserved. Counting Extended Irreducible Goppa Codes Wed, 12 Feb 2014 12:33:16 +0000 http://www.hindawi.com/journals/jdm/2014/871871/ We produce an upper bound on the number of extended irreducible Goppa codes over any finite field. John A. Ryan Copyright © 2014 John A. Ryan. All rights reserved. The Coarse Structure of the Representation Algebra of a Finite Monoid Thu, 30 Jan 2014 07:07:35 +0000 http://www.hindawi.com/journals/jdm/2014/529804/ Let be a monoid, and let be a commutative idempotent submonoid. We show that we can find a complete set of orthogonal idempotents of the monoid algebra of such that there is a basis of adapted to this set of idempotents which is in one-to-one correspondence with elements of the monoid. The basis graph describing the Peirce decomposition with respect to gives a coarse structure of the algebra, of which any complete set of primitive idempotents gives a refinement, and we give some criterion for this coarse structure to actually be a fine structure, which means that the nonzero elements of the monoid are in one-to-one correspondence with the vertices and arrows of the basis graph with respect to a set of primitive idempotents, with this basis graph being a canonical object. Mary Schaps Copyright © 2014 Mary Schaps. All rights reserved. Noncrossing Monochromatic Subtrees and Staircases in 0-1 Matrices Thu, 23 Jan 2014 16:22:51 +0000 http://www.hindawi.com/journals/jdm/2014/731519/ The following question is asked by the senior author (Gyárfás (2011)). What is the order of the largest monochromatic noncrossing subtree (caterpillar) that exists in every 2-coloring of the edges of a simple geometric ? We solve one particular problem asked by Gyárfás (2011): separate the Ramsey number of noncrossing trees from the Ramsey number of noncrossing double stars. We also reformulate the question as a Ramsey-type problem for 0-1 matrices and pose the following conjecture. Every 0-1 matrix contains zeros or ones, forming a staircase: a sequence which goes right in rows and down in columns, possibly skipping elements, but not at turning points. We prove this conjecture in some special cases and put forward some related problems as well. Siyuan Cai, Gillian Grindstaff, András Gyárfás, and Warren Shull Copyright © 2014 Siyuan Cai et al. All rights reserved. Combinatorial Interpretation of General Eulerian Numbers Thu, 02 Jan 2014 12:54:24 +0000 http://www.hindawi.com/journals/jdm/2014/870596/ Since the 1950s, mathematicians have successfully interpreted the traditional Eulerian numbers and -Eulerian numbers combinatorially. In this paper, the authors give a combinatorial interpretation to the general Eulerian numbers defined on general arithmetic progressions . Tingyao Xiong, Jonathan I. Hall, and Hung-Ping Tsao Copyright © 2014 Tingyao Xiong et al. All rights reserved. A Theory of Cartesian Product and Factorization of Circulant Graphs Thu, 26 Dec 2013 18:06:55 +0000 http://www.hindawi.com/journals/jdm/2013/163740/ We determine when the Cartesian product of two circulant graphs is also a circulant graph. This leads to a theory of factorization of circulant graphs. V. Vilfred Copyright © 2013 V. Vilfred. All rights reserved. Classification of Boolean Functions Where Affine Functions Are Uniformly Distributed Thu, 31 Oct 2013 15:18:06 +0000 http://www.hindawi.com/journals/jdm/2013/270424/ The present paper on classification of -variable Boolean functions highlights the process of classification in a coherent way such that each class contains a single affine Boolean function. Two unique and different methods have been devised for this classification. The first one is a recursive procedure that uses the Cartesian product of sets starting from the set of one variable Boolean functions. In the second method, the classification is done by changing some predefined bit positions with respect to the affine function belonging to that class. The bit positions which are changing also provide us information concerning the size and symmetry properties of the classes/subclasses in such a way that the members of classes/subclasses satisfy certain similar properties. Ranjeet Kumar Rout, Pabitra Pal Choudhury, and Sudhakar Sahoo Copyright © 2013 Ranjeet Kumar Rout et al. All rights reserved. On a Property of a Three-Dimensional Matrix Wed, 30 Oct 2013 12:01:07 +0000 http://www.hindawi.com/journals/jdm/2013/797249/ Let be the symmetrical group acting on the set and . Consider the set The main result of this paper is the following theorem. If the number of set entries is more than , then there exist entries such that , , and . The application of this theorem to the three-dimensional assignment problem is considered. David Blokh Copyright © 2013 David Blokh. All rights reserved. -Analogues of Symbolic Operators Sun, 14 Jul 2013 09:34:08 +0000 http://www.hindawi.com/journals/jdm/2013/487546/ Here presented are -extensions of several linear operators including a novel -analogue of the derivative operator . Some -analogues of the symbolic substitution rules given by He et al., 2007, are obtained. As sample applications, we show how these -substitution rules may be used to construct symbolic summation and series transformation formulas, including -analogues of the classical Euler transformations for accelerating the convergence of alternating series. Michael J. Dancs and Tian-Xiao He Copyright © 2013 Michael J. Dancs and Tian-Xiao He. All rights reserved. Terminal Hosoya Polynomial of Line Graphs Thu, 13 Jun 2013 09:28:16 +0000 http://www.hindawi.com/journals/jdm/2013/857908/ The terminal Hosoya polynomial of a graph is defined as , where is the number of pairs of pendant vertices of that are at distance . In this paper we obtain terminal Hosoya polynomial of line graphs. H. S. Ramane, A. B. Ganagi, K. P. Narayankar, and S. S. Shirkol Copyright © 2013 H. S. Ramane et al. All rights reserved. Equivalence of Right Infinite Words Mon, 15 Apr 2013 14:17:01 +0000 http://www.hindawi.com/journals/jdm/2013/219291/ Closure properties of some classes of right infinite words have been studied extensively; we are interested in the general algebraic structure of right infinite words. We investigate preorder of morphism invariant classes and show that it is not a semilattice. Liga Kulesa Copyright © 2013 Liga Kulesa. All rights reserved. Decomposition of Graphs into Paths and Cycles Sun, 14 Apr 2013 08:45:07 +0000 http://www.hindawi.com/journals/jdm/2013/721051/ A decomposition of a graph is a collection of edge-disjoint subgraphs of such that every edge of belongs to exactly one . If each is a path or a cycle in , then is called a path decomposition of . If each is a path in , then is called an acyclic path decomposition of . The minimum cardinality of a path decomposition (acyclic path decomposition) of is called the path decomposition number (acyclic path decomposition number) of and is denoted by () (()). In this paper we initiate a study of the parameter and determine the value of for some standard graphs. Further, we obtain some bounds for and characterize graphs attaining the bounds. We also prove that the difference between the parameters and can be made arbitrarily large. S. Arumugam, I. Sahul Hamid, and V. M. Abraham Copyright © 2013 S. Arumugam et al. All rights reserved. A Note on the Adversary Degree Associated Reconstruction Number of Graphs Sun, 31 Mar 2013 13:56:48 +0000 http://www.hindawi.com/journals/jdm/2013/808105/ A vertex-deleted subgraph of a graph is called a card of . A card of with which the degree of the deleted vertex is also given is called a degree associated card (or dacard) of . The degree associated reconstruction number drn () of a graph is the size of the smallest collection of dacards of that uniquely determines . The adversary degree associated reconstruction number of a graph , adrn(), is the minimum number such that every collection of dacards of that uniquely determines . In this paper, we show that adrn of wheels and complete bipartite graphs on at least 4 vertices is 2 or 3. S. Monikandan, S. Sundar Raj, C. Jayasekaran, and A. P. Santhakumaran Copyright © 2013 S. Monikandan et al. All rights reserved. Efficient Prime Counting and the Chebyshev Primes Mon, 25 Mar 2013 08:24:56 +0000 http://www.hindawi.com/journals/jdm/2013/491627/ The function where is the logarithm integral and the number of primes up to is well known to be positive up to the (very large) Skewes' number. Likewise, according to Robin's work, the functions and , where and are Chebyshev summatory functions, are positive if and only if Riemann hypothesis (RH) holds. One introduces the jump function at primes and one investigates , , and . In particular, , and for . Besides, for any odd , an infinite set of the so-called Chebyshev primes. In the context of RH, we introduce the so-called Riemann primes as champions of the function (or of the function ). Finally, we find a good prime counting function , that is found to be much better than the standard Riemann prime counting function. Michel Planat and Patrick Solé Copyright © 2013 Michel Planat and Patrick Solé. All rights reserved. -Pascal and -Wronskian Matrices with Implications to -Appell Polynomials Wed, 20 Mar 2013 18:22:29 +0000 http://www.hindawi.com/journals/jdm/2013/450481/ We introduce a -deformation of the Yang and Youn matrix approach for Appell polynomials. This will lead to a powerful machinery for producing new and old formulas for -Appell polynomials, and in particular for -Bernoulli and -Euler polynomials. Furthermore, the --polynomial, anticipated by Ward, can be expressed as a sum of products of -Bernoulli and -Euler polynomials. The pseudo -Appell polynomials, which are first presented in this paper, enable multiple -analogues of the Yang and Youn formulas. The generalized -Pascal functional matrix, the -Wronskian vector of a function, and the vector of -Appell polynomials together with the -deformed matrix multiplication from the authors recent article are the main ingredients in the process. Beyond these results, we give a characterization of -Appell numbers, improving on Al-Salam 1967. Finally, we find a -difference equation for the -Appell polynomial of degree . Thomas Ernst Copyright © 2013 Thomas Ernst. All rights reserved.