ISRN Discrete Mathematics The latest articles from Hindawi Publishing Corporation © 2014 , Hindawi Publishing Corporation . All rights reserved. The Graph of Equivalence Classes of Zero Divisors Thu, 08 May 2014 10:09:48 +0000 We introduce a graph of equivalence classes of zero divisors of a meet semilattice with 0. The set of vertices of are the equivalence classes of nonzero zero divisors of and two vertices and are adjacent if and only if . It is proved that is connected and either it contains a cycle of length 3 or . It is known that two Boolean lattices and have isomorphic zero divisor graphs if and only if . This result is extended to the class of SSC meet semilattices. Finally, we show that Beck's Conjecture is true for . Vinayak Joshi, B. N. Waphare, and H. Y. Pourali Copyright © 2014 Vinayak Joshi et al. All rights reserved. -Step Sum and -Step Gap Fibonacci Sequence Wed, 09 Apr 2014 12:21:49 +0000 For two given integers , , we introduce the -step sumand -step gap Fibonacci sequence by presenting a recurrence formula that generates the th term as the sum of successive previous terms starting the sum at the th previous term. Known sequences, like Fibonacci, tribonacci, tetranacci, and Padovan sequences, are derived for specific values of , . Two limiting properties concerning the terms of the sequence are presented. The limits are related to the spectral radius of the associated -matrix. Maria Adam and Nicholas Assimakis Copyright © 2014 Maria Adam and Nicholas Assimakis. All rights reserved. Edge Domination in Some Path and Cycle Related Graphs Thu, 13 Mar 2014 13:50:55 +0000 For a graph , a subset of is called an edge dominating set of if every edge not in is adjacent to some edge in . The edge domination number of is the minimum cardinality taken over all edge dominating sets of . Here, we determine the edge domination number for shadow graphs, middle graphs, and total graphs of paths and cycles. S. K. Vaidya and R. M. Pandit Copyright © 2014 S. K. Vaidya and R. M. Pandit. All rights reserved. Mathematical Morphology on Hypergraphs Using Vertex-Hyperedge Correspondence Thu, 13 Mar 2014 08:40:18 +0000 The focus of this paper is to develop computationally efficient mathematical morphology operators on hypergraphs. To this aim we consider lattice structures on hypergraphs on which we build morphological operators. We develop a pair of dual adjunctions between the vertex set and the hyperedge set of a hypergraph , by defining a vertex-hyperedge correspondence. This allows us to recover the classical notion of a dilation/erosion of a subset of vertices and to extend it to subhypergraphs of . This paper also studies the concept of morphological adjunction on hypergraphs for which both the input and the output are hypergraphs. Bino Sebastian, A. Unnikrishnan, Kannan Balakrishnan, and P. B. Ramkumar Copyright © 2014 Bino Sebastian et al. All rights reserved. A Note on Closed-Form Representation of Fibonacci Numbers Using Fibonacci Trees Wed, 12 Feb 2014 07:10:49 +0000 We give a new representation of the Fibonacci numbers. This is achieved using Fibonacci trees. With the help of this representation, the th Fibonacci number can be calculated without having any knowledge about the previous Fibonacci numbers. Indhumathi Raman Copyright © 2014 Indhumathi Raman. All rights reserved. A Bijection for Tricellular Maps Sun, 08 Dec 2013 11:14:33 +0000 We give a bijective proof for a relation between unicellular, bicellular, and tricellular maps. These maps represent cell complexes of orientable surfaces having one, two, or three boundary components. The relation can formally be obtained using matrix theory (Dyson, 1949) employing the Schwinger-Dyson equation (Schwinger, 1951). In this paper we present a bijective proof of the corresponding coefficient equation. Our result is a bijection that transforms a unicellular map of genus into unicellular, bicellular or tricellular maps of strictly lower genera. The bijection employs edge cutting, edge contraction, and edge deletion. Hillary S. W. Han and Christian M. Reidys Copyright © 2013 Hillary S. W. Han and Christian M. Reidys. All rights reserved. Graphs Whose Certain Polynomials Have Few Distinct Roots Thu, 12 Sep 2013 18:45:49 +0000 Let be a simple graph. Graph polynomials are a well-developed area useful for analyzing properties of graphs. We consider domination polynomial, matching polynomial, and edge cover polynomial of . Graphs which their polynomials have few roots can sometimes give surprising information about the structure of the graph. This paper is primarily a survey of graphs whose domination polynomial, matching polynomial, and edge cover polynomial have few distinct roots. In addition, some new unpublished results and questions are concluded. Saeid Alikhani Copyright © 2013 Saeid Alikhani. All rights reserved. Removable Cycles Avoiding Two Connected Subgraphs Wed, 27 Mar 2013 14:19:45 +0000 We provide a sufficient condition for the existence of a cycle in a connected graph which is edge-disjoint from two connected subgraphs and of such that is connected. Y. M. Borse and B. N. Waphare Copyright © 2013 Y. M. Borse and B. N. Waphare. All rights reserved. Some Properties of a Sequence Similar to Generalized Euler Numbers Mon, 11 Mar 2013 15:49:37 +0000 We introduce the sequence given by generating function and establish some explicit formulas for the sequence . Several identities involving the sequence , Stirling numbers, Euler polynomials, and the central factorial numbers are also presented. Haiqing Wang and Guodong Liu Copyright © 2013 Haiqing Wang and Guodong Liu. All rights reserved. An Asymptotic Formula for -Bell Numbers with Real Arguments Wed, 13 Feb 2013 08:36:36 +0000 The -Bell numbers are generalized using the concept of the Hankel contour. Some properties parallel to those of the ordinary Bell numbers are established. Moreover, an asymptotic approximation for -Bell numbers with real arguments is obtained. Cristina B. Corcino and Roberto B. Corcino Copyright © 2013 Cristina B. Corcino and Roberto B. Corcino. All rights reserved. Pairwise Balanced Design of Order and 2-Fold System of Order Thu, 27 Dec 2012 11:02:33 +0000 In the Steiner triple system, Bose (1939) constructed the design for and later on Skolem (1958) constructed the same for . In the literature we found a pairwise balanced design (PBD) for . We also found the 2-fold triple system of the orders 3n and . In this paper, we construct a PBD for and a 2-fold system of the order . The second construction completes the 2-fold system for all . Manjusri Basu, Debabrata Kumar Ghosh, and Satya Bagchi Copyright © 2012 Manjusri Basu et al. All rights reserved. Some Results Involving the Splitting Operation on Binary Matroids Sun, 16 Dec 2012 15:11:04 +0000 We obtain some results concerning the planarity and graphicness of the splitting matroids. Further, we explore the effect of splitting operation on the sum of two matroids. Naiyer Pirouz and Maruti Mukinda Shikare Copyright © 2012 Naiyer Pirouz and Maruti Mukinda Shikare. All rights reserved. Bipartite Graphs Related to Mutually Disjoint S-Permutation Matrices Sun, 09 Dec 2012 14:45:20 +0000 Some numerical characteristics of bipartite graphs in relation to the problem of finding all disjoint pairs of S-permutation matrices in the general case are discussed in this paper. All bipartite graphs of the type , where or , are provided. The cardinality of the sets of mutually disjoint S-permutation matrices in both the and cases is calculated. Krasimir Yordzhev Copyright © 2012 Krasimir Yordzhev. All rights reserved. On the Adjacent Cycle Derangements Sun, 02 Dec 2012 15:52:24 +0000 A derangement, that is, a permutation without fixed points, of a finite set is said to be an adjacent cycle when all its cycles are formed by a consecutive set of integers. In this paper we determine enumerative properties of these permutations using analytical and bijective proofs. Moreover a combinatorial interpretation in terms of linear species is provided. Finally we define and investigate the case of the adjacent cycle derangements of a multiset. Luisa de Francesco Albasini and Norma Zagaglia Salvi Copyright © 2012 Luisa de Francesco Albasini and Norma Zagaglia Salvi. All rights reserved. Spider Covers and Their Applications Wed, 28 Nov 2012 14:41:28 +0000 We introduce two new combinatorial optimization problems: the Maximum Spider Problem and the Spider Cover Problem; we study their approximability and illustrate their applications. In these problems we are given a directed graph , a distinguished vertex , and a family D of subsets of vertices. A spider centered at vertex s is a collection of arc-disjoint paths all starting at s but ending into pairwise distinct vertices. We say that a spider covers a subset of vertices X if at least one of the endpoints of the paths constituting the spider other than s belongs to X. In the Maximum Spider Problem the goal is to find a spider centered at s that covers the maximum number of elements of the family D. Conversely, the Spider Cover Problem consists of finding the minimum number of spiders centered at s that covers all subsets in D. We motivate the study of the Maximum Spider and Spider Cover Problems by pointing out a variety of applications. We show that a natural greedy algorithm gives a 2-approximation algorithm for the Maximum Spider Problem and a -approximation algorithm for the Spider Cover Problem. Filomena De Santis, Luisa Gargano, Mikael Hammar, Alberto Negro, and Ugo Vaccaro Copyright © 2012 Filomena De Santis et al. All rights reserved. Splitting Lemma for 2-Connected Graphs Sun, 25 Nov 2012 14:04:58 +0000 Using a splitting operation and a splitting lemma for connected graphs, Fleischner characterized connected Eulerian graphs. In this paper, we obtain a splitting lemma for 2-connected graphs and characterize 2-connected Eulerian graphs. As a consequence, we characterize connected graphic Eulerian matroids. Y. M. Borse Copyright © 2012 Y. M. Borse. All rights reserved. Median Sets and Median Number of a Graph Wed, 21 Nov 2012 11:53:59 +0000 A profile is a finite sequence of vertices of a graph. The set of all vertices of the graph which minimises the sum of the distances to the vertices of the profile is the median of the profile. Any subset of the vertex set such that it is the median of some profile is called a median set. The number of median sets of a graph is defined to be the median number of the graph. In this paper, we identify the median sets of various classes of graphs such as , for , and wheel graph and so forth. The median numbers of these graphs and hypercubes are found out, and an upper bound for the median number of even cycles is established. We also express the median number of a product graph in terms of the median number of their factors. R. Ram Kumar and B. Kannan Copyright © 2012 R. Ram Kumar and B. Kannan. All rights reserved. Some New Results on Global Dominating Sets Mon, 05 Nov 2012 08:15:53 +0000 A dominating set is called a global dominating set if it is a dominating set of a graph and its complement . A natural question arises: are there any graphs for which it is possible to relate the domination number and the global domination number? We have found an affirmative answer to this question and obtained some graphs having such characteristic. S. K. Vaidya and R. M. Pandit Copyright © 2012 S. K. Vaidya and R. M. Pandit. All rights reserved. A -Analogue of Rucinski-Voigt Numbers Thu, 18 Oct 2012 15:11:14 +0000 A -analogue of Rucinski-Voigt numbers is defined by means of a recurrence relation, and some properties including the orthogonality and inverse relations with the -analogue of the limit of the differences of the generalized factorial are obtained. Roberto B. Corcino and Charles B. Montero Copyright © 2012 Roberto B. Corcino and Charles B. Montero. All rights reserved. Global Exponential Stability of Discrete-Time Multidirectional Associative Memory Neural Network with Variable Delays Tue, 16 Oct 2012 11:07:04 +0000 A discrete-time multidirectional associative memory neural networks model with varying time delays is formulated by employing the semidiscretization method. A sufficient condition for the existence of an equilibrium point is given. By calculating difference and using inequality technique, a sufficient condition for the global exponential stability of the equilibrium point is obtained. The results are helpful to design global exponentially stable multidirectional associative memory neural networks. An example is given to illustrate the effectiveness of the results. Min Wang, Tiejun Zhou, and Xiaolan Zhang Copyright © 2012 Min Wang et al. All rights reserved. Bipancyclic Properties of Faulty Hypercubes Sun, 14 Oct 2012 15:05:06 +0000 A bipartite graph is bipancyclic if it contains cycles of every even length from 4 to and edge bipancyclic if every edge lies on a cycle of every even length from 4 to . Let denote the -dimensional hypercube. Let be a subset of such that can be decomposed into two parts and , where is a union of disjoint adjacent pairs of , and consists of edges. We prove that is bipancyclic if . Moreover, is edge bipancyclic if with . Chun-Nan Hung and Min-Kun Hsiao Copyright © 2012 Chun-Nan Hung and Min-Kun Hsiao. All rights reserved. Uniqueness of the Infinite Component for Percolation on a Hierarchical Lattice Tue, 11 Sep 2012 08:36:37 +0000 We study a long-range percolation in the hierarchical lattice Ω𝑁 of order 𝑁 where probability of connection between two nodes separated by distance π‘˜ is of the form min{π›Όπ›½βˆ’π‘˜,1}, 𝛼β‰₯0 and 𝛽>0. We show the uniqueness of the infinite component for this model. Yilun Shang Copyright © 2012 Yilun Shang. All rights reserved. Explicit Evaluations of Cubic and Quartic Theta-Functions Thu, 31 May 2012 13:11:50 +0000 We find explicit values of cubic and quartic theta-functions and their quotients by parameterizations. In the process, we also find some transformation formulas of these theta-functions. Nipen Saikia Copyright © 2012 Nipen Saikia. All rights reserved. Weighted Maximum-Clique Transversal Sets of Graphs Thu, 26 Jan 2012 16:59:54 +0000 A maximum-clique transversal set of a graph G is a subset of vertices intersecting all maximum cliques of G. The maximum-clique transversal set problem is to find a maximum-clique transversal set of G of minimum cardinality. Motivated by the placement of transmitters for cellular telephones, Chang, Kloks, and Lee introduced the concept of maximum-clique transversal sets on graphs in 2001. In this paper, we study the weighted version of the maximum-clique transversal set problem for split graphs, balanced graphs, strongly chordal graph, Helly circular-arc graphs, comparability graphs, distance-hereditary graphs, and graphs of bounded treewidth. Chuan-Min Lee Copyright Β© 2011 Chuan-Min Lee. All rights reserved. π‘˜-Tuple Total Domination in Complementary Prisms Wed, 18 Jan 2012 13:44:27 +0000 Let π‘˜ be a positive integer, and let 𝐺 be a graph with minimum degree at least π‘˜. In their study (2010), Henning and Kazemi defined the π‘˜-tuple total domination number π›ΎΓ—π‘˜,𝑑(𝐺) of 𝐺 as the minimum cardinality of a π‘˜-tuple total dominating set of 𝐺, which is a vertex set such that every vertex of 𝐺 is adjacent to at least π‘˜ vertices in it. If 𝐺 is the complement of 𝐺, the complementary prism 𝐺𝐺 of 𝐺 is the graph formed from the disjoint union of 𝐺 and 𝐺 by adding the edges of a perfect matching between the corresponding vertices of 𝐺 and 𝐺. In this paper, we extend some of the results of Haynes et al. (2009) for the π‘˜-tuple total domination number and also obtain some other new results. Also we find the π‘˜-tuple total domination number of the complementary prism of a cycle, a path, or a complete multipartite graph. Adel P. Kazemi Copyright © 2011 Adel P. Kazemi. All rights reserved. On the Spectrum of Threshold Graphs Tue, 17 Jan 2012 11:02:00 +0000 The antiregular connected graph on π‘Ÿ vertices is defined as the connected graph whose vertex degrees take the values of π‘Ÿβˆ’1 distinct positive integers. We explore the spectrum of its adjacency matrix and show common properties with those of connected threshold graphs, having an equitable partition with a minimal number π‘Ÿ of parts. Structural and combinatorial properties can be deduced for related classes of graphs and in particular for the minimal configurations in the class of singular graphs. Irene Sciriha and Stephanie Farrugia Copyright Β© 2011 Irene Sciriha and Stephanie Farrugia. All rights reserved. Exponential Stability for Discrete-Time Stochastic BAM Neural Networks with Discrete and Distributed Delays Mon, 16 Jan 2012 08:32:52 +0000 This paper deals with the stability analysis problem for a class of discrete-time stochastic BAM neural networks with discrete and distributed time-varying delays. By constructing a suitable Lyapunov-Krasovskii functional and employing M-matrix theory, we find some sufficient conditions ensuring the global exponential stability of the equilibrium point for stochastic BAM neural networks with time-varying delays. The conditions obtained here are expressed in terms of LMIs whose feasibility can be easily checked by MATLAB LMI Control toolbox. A numerical example is presented to show the effectiveness of the derived LMI-based stability conditions. R. Raja, R. Sakthivel, and S. Marshal Anthoni Copyright Β© 2011 R. Raja et al. All rights reserved. Equivalence between Hypergraph Convexities Sun, 15 Jan 2012 08:47:20 +0000 Let 𝐺 be a connected graph on 𝑉. A subset 𝑋 of 𝑉 is all-paths convex (or ap-convex) if 𝑋 contains each vertex on every path joining two vertices in 𝑋 and is monophonically convex (or π‘š-convex) if 𝑋 contains each vertex on every chordless path joining two vertices in 𝑋. First of all, we prove that ap-convexity and π‘š-convexity coincide in 𝐺 if and only if 𝐺 is a tree. Next, in order to generalize this result to a connected hypergraph 𝐻, in addition to the hypergraph versions of ap-convexity and π‘š-convexity, we consider canonical convexity (or 𝑐-convexity) and simple-path convexity (or sp-convexity) for which it is well known that π‘š-convexity is finer than both 𝑐-convexity and sp-convexity and sp-convexity is finer than ap-convexity. After proving sp-convexity is coarser than 𝑐-convexity, we characterize the hypergraphs in which each pair of the four convexities above is equivalent. As a result, we obtain a convexity-theoretic characterization of Berge-acyclic hypergraphs and of 𝛾-acyclic hypergraphs. Francesco M. Malvestuto, Mauro Mezzini, and Marina Moscarini Copyright © 2011 Francesco M. Malvestuto et al. All rights reserved. Notes on the Union of Weakly Primary Submodules Wed, 28 Dec 2011 12:03:23 +0000 Let 𝑅 be a commutative ring with identity, and let 𝑀 be an 𝑅-module. A proper submodule 𝑁 of 𝑀 is said to be weakly primary if 0β‰ π‘Ÿπ‘šβˆˆπ‘ for π‘Ÿβˆˆπ‘… and π‘šβˆˆπ‘€, which implies that either π‘šβˆˆπ‘ or π‘Ÿπ‘›π‘€βŠ†π‘ for some positive integer 𝑛. In this paper, we study weakly primary submodules, and we investigate the union of weakly primary submodules of 𝑅-modules. Peyman Ghiasvand and Farkhonde Farzalipour Copyright © 2011 Peyman Ghiasvand and Farkhonde Farzalipour. All rights reserved. The 𝐿-Total Graph of an 𝐿-Module Wed, 07 Dec 2011 13:55:03 +0000 Let 𝐿 be a complete lattice. We introduce and investigate the 𝐿-total graph of an 𝐿-module over an 𝐿-commutative ring. The main purpose of this paper is to extend the definition and results given in (Anderson and Badawi, 2008) to more generalize the 𝐿-total graph of an 𝐿-module case. Reza Ebrahimi Atani Copyright © 2011 Reza Ebrahimi Atani. All rights reserved.