ISRN Discrete Mathematics
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Step Sum and Step Gap Fibonacci Sequence
Wed, 09 Apr 2014 12:21:49 +0000
http://www.hindawi.com/journals/isrn.discrete.mathematics/2014/374902/
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
http://www.hindawi.com/journals/isrn.discrete.mathematics/2014/975812/
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 VertexHyperedge Correspondence
Thu, 13 Mar 2014 08:40:18 +0000
http://www.hindawi.com/journals/isrn.discrete.mathematics/2014/436419/
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 vertexhyperedge 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 ClosedForm Representation of Fibonacci Numbers Using Fibonacci Trees
Wed, 12 Feb 2014 07:10:49 +0000
http://www.hindawi.com/journals/isrn.discrete.mathematics/2014/132925/
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
http://www.hindawi.com/journals/isrn.discrete.mathematics/2013/712431/
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 SchwingerDyson 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
http://www.hindawi.com/journals/isrn.discrete.mathematics/2013/195818/
Let be a simple graph. Graph polynomials are a welldeveloped 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
http://www.hindawi.com/journals/isrn.discrete.mathematics/2013/164535/
We provide a sufficient condition for the existence of a cycle in a connected graph which is edgedisjoint 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
http://www.hindawi.com/journals/isrn.discrete.mathematics/2013/810245/
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
http://www.hindawi.com/journals/isrn.discrete.mathematics/2013/274697/
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 2Fold System of Order
Thu, 27 Dec 2012 11:02:33 +0000
http://www.hindawi.com/journals/isrn.discrete.mathematics/2012/251457/
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 2fold triple system of the orders 3n and . In this paper, we construct a PBD for and a 2fold system of the order . The second construction completes the 2fold 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
http://www.hindawi.com/journals/isrn.discrete.mathematics/2012/406147/
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 SPermutation Matrices
Sun, 09 Dec 2012 14:45:20 +0000
http://www.hindawi.com/journals/isrn.discrete.mathematics/2012/384068/
Some numerical characteristics of bipartite graphs in relation to the problem of finding all disjoint pairs of Spermutation 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 Spermutation 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
http://www.hindawi.com/journals/isrn.discrete.mathematics/2012/340357/
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
http://www.hindawi.com/journals/isrn.discrete.mathematics/2012/347430/
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 arcdisjoint 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 2approximation 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 2Connected Graphs
Sun, 25 Nov 2012 14:04:58 +0000
http://www.hindawi.com/journals/isrn.discrete.mathematics/2012/850538/
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 2connected graphs and characterize 2connected 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
http://www.hindawi.com/journals/isrn.discrete.mathematics/2012/583671/
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
http://www.hindawi.com/journals/isrn.discrete.mathematics/2012/852129/
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 RucinskiVoigt Numbers
Thu, 18 Oct 2012 15:11:14 +0000
http://www.hindawi.com/journals/isrn.discrete.mathematics/2012/592818/
A analogue of RucinskiVoigt 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 DiscreteTime Multidirectional Associative Memory Neural Network with Variable Delays
Tue, 16 Oct 2012 11:07:04 +0000
http://www.hindawi.com/journals/isrn.discrete.mathematics/2012/831715/
A discretetime 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
http://www.hindawi.com/journals/isrn.discrete.mathematics/2012/308595/
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 .
ChunNan Hung and MinKun Hsiao
Copyright © 2012 ChunNan Hung and MinKun Hsiao. All rights reserved.

Uniqueness of the Infinite Component for Percolation on a Hierarchical Lattice
Tue, 11 Sep 2012 08:36:37 +0000
http://www.hindawi.com/journals/isrn.discrete.mathematics/2012/758721/
We study a longrange percolation in the hierarchical lattice Ξ©π of order π where probability of connection between two nodes separated by distance π is of the form min{πΌπ½βπ,1}, πΌβ₯0 and π½>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 ThetaFunctions
Thu, 31 May 2012 13:11:50 +0000
http://www.hindawi.com/journals/isrn.discrete.mathematics/2012/956594/
We find explicit values of cubic and quartic thetafunctions and their quotients by parameterizations. In the process, we also find some transformation formulas of these thetafunctions.
Nipen Saikia
Copyright © 2012 Nipen Saikia. All rights reserved.

Weighted MaximumClique Transversal Sets of Graphs
Thu, 26 Jan 2012 16:59:54 +0000
http://www.hindawi.com/journals/isrn.discrete.mathematics/2011/540834/
A maximumclique transversal set of a graph G is a subset of vertices intersecting all maximum cliques of G. The maximumclique transversal set problem is to find a maximumclique transversal set of G of minimum cardinality. Motivated by the placement of transmitters for cellular telephones, Chang, Kloks, and Lee introduced the concept of maximumclique transversal sets on graphs in 2001. In this paper, we study the weighted version of the maximumclique transversal set problem for split graphs, balanced graphs, strongly chordal graph, Helly circulararc graphs, comparability graphs, distancehereditary graphs, and graphs of bounded treewidth.
ChuanMin Lee
Copyright Β© 2011 ChuanMin Lee. All rights reserved.

πTuple Total Domination in Complementary Prisms
Wed, 18 Jan 2012 13:44:27 +0000
http://www.hindawi.com/journals/isrn.discrete.mathematics/2011/681274/
Let π be a positive integer, and let πΊ be a graph with minimum degree at least π. In their study (2010), Henning and Kazemi defined the πtuple total domination number πΎΓπ,π‘(πΊ) of πΊ as the minimum cardinality of a πtuple total dominating set of πΊ, which is a vertex set such that every vertex of πΊ is adjacent to at least π vertices in it. If πΊ is the complement of πΊ, the complementary prism πΊπΊ of πΊ is the graph formed from the disjoint union of πΊ and πΊ by adding the edges of a perfect matching between the corresponding vertices of πΊ and πΊ. In this paper, we extend
some of the results of Haynes et al. (2009) for the πtuple total domination number
and also obtain some other new results. Also we find the πtuple 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
http://www.hindawi.com/journals/isrn.discrete.mathematics/2011/108509/
The antiregular connected graph on π vertices is defined as the connected graph whose vertex degrees take the values of πβ1 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 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 © 2011 Irene Sciriha and Stephanie Farrugia. All rights reserved.

Exponential Stability for DiscreteTime Stochastic BAM Neural Networks with Discrete and Distributed Delays
Mon, 16 Jan 2012 08:32:52 +0000
http://www.hindawi.com/journals/isrn.discrete.mathematics/2011/153409/
This paper deals with the stability analysis problem for a class of discretetime stochastic
BAM neural networks with discrete and distributed timevarying delays. By constructing a suitable
LyapunovKrasovskii functional and employing Mmatrix theory, we find some sufficient
conditions ensuring the global exponential stability of the equilibrium point for stochastic BAM
neural networks with timevarying 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 LMIbased stability conditions.
R. Raja, R. Sakthivel, and S. Marshal Anthoni
Copyright © 2011 R. Raja et al. All rights reserved.

Equivalence between Hypergraph Convexities
Sun, 15 Jan 2012 08:47:20 +0000
http://www.hindawi.com/journals/isrn.discrete.mathematics/2011/806193/
Let πΊ be a connected graph on π. A subset π of π is allpaths convex (or apconvex) if π contains each vertex on every path joining two vertices in π and is monophonically convex (or πconvex) if π contains each vertex on every chordless path joining two vertices in π. First of all, we prove that apconvexity and πconvexity coincide in πΊ if and only if πΊ is a tree. Next, in order to generalize this result to a connected hypergraph π», in addition to the hypergraph versions of apconvexity and πconvexity, we consider canonical convexity (or πconvexity) and simplepath convexity (or spconvexity) for which it is well known that πconvexity is finer than both πconvexity and spconvexity and spconvexity is finer than apconvexity. After proving spconvexity is coarser than πconvexity, we characterize the hypergraphs in which each pair of the four convexities above is equivalent. As a result, we obtain a convexitytheoretic characterization of Bergeacyclic hypergraphs and of πΎacyclic 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
http://www.hindawi.com/journals/isrn.discrete.mathematics/2011/939687/
Let π
be a commutative ring with identity, and let π be an π
module. A proper submodule π of π is said to be weakly primary if 0β ππβπ for πβπ
and πβπ, which implies that either πβπ or πππβπ for some positive integer π. In this paper, we study weakly primary submodules, and we investigate the union of weakly primary submodules of π
modules.
Peyman Ghiasvand and Farkhonde Farzalipour
Copyright © 2011 Peyman Ghiasvand and Farkhonde Farzalipour. All rights reserved.

The πΏTotal Graph of an πΏModule
Wed, 07 Dec 2011 13:55:03 +0000
http://www.hindawi.com/journals/isrn.discrete.mathematics/2011/491936/
Let πΏ be a complete lattice. We introduce and investigate
the πΏtotal graph of an πΏmodule over an πΏcommutative ring. The main
purpose of this paper is to extend the definition and results given in (Anderson and Badawi, 2008) to
more generalize the πΏtotal graph of an πΏmodule case.
Reza Ebrahimi Atani
Copyright © 2011 Reza Ebrahimi Atani. All rights reserved.

An Efficient Algorithm to Solve the Conditional Covering Problem on Trapezoid Graphs
Thu, 17 Nov 2011 15:11:07 +0000
http://www.hindawi.com/journals/isrn.discrete.mathematics/2011/213084/
Let πΊ=(π,πΈ) be a simple connected undirected graph. Each vertex π£βπ has a cost π(π£) and provides a positive coverage radius π
(π£). A distance πuv is associated with each edge {π’,π£}βπΈ, and π(π’,π£) is the shortest distance between every pair of vertices π’,π£βπ. A vertex π£ can cover all vertices that lie within the distance π
(π£), except the vertex itself. The conditional covering problem is to minimize the sum of the costs required to cover all the vertices in πΊ. This problem is NPcomplete for general graphs, even it remains NPcomplete for chordal graphs. In this paper, an π(π2) time algorithm to solve a special case of the problem in a trapezoid graph is proposed, where π is the number of vertices of the graph. In this special
case, πuv=1 for every edge {π’,π£}βπΈ, π(π£)=π for every π£βπ(πΊ), and π
(π£)=π
, an integer >1, for every π£βπ(πΊ). A new data structure on trapezoid graphs is used to solve the problem.
Akul Rana, Anita Pal, and Madhumangal Pal
Copyright © 2011 Akul Rana et al. All rights reserved.