ISSN: 1815-4816

Volume 9, Issue 2

Volume 9, Issue 2, Autumn 2012, Page 13-257

A Combined Cubic and Novel Line Search CG-Algorithm

Abbas Y. Al-Bayati; Hamsa Th. Chilmerane

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 13-24
DOI: 10.33899/csmj.2012.163697

        In this paper a new line search technique is investigated. It uses (cubic and novel) line searches in the standard CG-algorithm for unconstrained optimization. Applying our new modified version on CG-method shows that, it is too effective when compared with other established algorithms, in this paper, to solve standard unconstrained optimization problems.

A New Preconditioned Inexact Line-Search Technique for Unconstrained Optimization

Abbas Y. Al-Bayati; Ivan S. Latif

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 25-39
DOI: 10.33899/csmj.2012.163698

In this paper, we study the global convergence properties of the new class of preconditioned conjugate gradient descent algorithm, when applied to convex objective non-linear unconstrained optimization functions.
We assume that a new inexact line search rule which is similar to the Armijo line-search rule is used. It's an estimation formula to choose a large step-size at each iteration and use the same formula to find the direction search. A new preconditioned conjugate gradient direction search is used to replace the conjugate gradient descent direction of ZIR-algorithm. Numerical results on twenty five well-know test functions with various dimensions show that the new inexact line-search and the new preconditioned conjugate gradient search directions are efficient for solving unconstrained nonlinear optimization problem in many situations.

Romberg Algorithm to Solve a System of Nonlinear Fredholm Integral Equations of Second Kind

Abbas Y. Al-Bayati; Rostam K. Saeed; Kawa M. Aziz

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 41-52
DOI: 10.33899/csmj.2012.163699

In this paper, we use Romberg algorithm, to find an approximation solution for a system of nonlinear Fredholm integral equations of the second kind. A comparison between this approximations and exact solutions for four numerical examples are given, depending on the least-square error and running computer time. Our numerical results are obtained by MATLAB 7.0 program and for a system of rank 2×2, 3×3 on Personal Computer P4.

On n-Weakly Regular Rings

Raida D. Mahammod; Mohammed Th. Al-Neimi

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 53-59
DOI: 10.33899/csmj.2012.163700

As a generalization of right weakly regular rings, we introduce the notion of right n-weakly regular rings, i.e. for all aN(R), aaRaR. In this paper, first  give  various properties of right n-weakly regular rings. Also, we study the relation between such rings and reduced rings by adding some types of rings, such as NCI, MC2 and SNF rings.

On m-Regular Rings

Zubayda M. Ibraheem; Maha F. Khalaf

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 61-70
DOI: 10.33899/csmj.2012.163701

As a generalization of regular rings, we introduce the notion, of m-regular rings, that is for all , there is a fixed positive integer m such that  is a Von-Neumann regular element. Some characterization and basic properties of these rings will be given. Also, we study the relation-ship between them and Von-Neumann regular rings, -regular rings, reduced rings, locally rings, uniform rings and 2-primal rings.

Local Existence Theorem of Fractional Differential Equations in Lp Space

Joseph Gh. Abdulahad; Shayma A. Murad

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 71-78
DOI: 10.33899/csmj.2012.163702

We proved the existence of P-integrable solution in -space, where  for the  fractional differential equation which has the form:
with boundary condition
where is the Caputo fractional derivative,  and c are positive constants with . The contraction mapping principle has been used to establish our main result.

Alternating Direction Implicit Method for Solving Parabolic Partial Differential Equations in Three Dimensions

Abdulghafor M. Al-Rozbayani; Mahmood H. Yahya

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 79-97
DOI: 10.33899/csmj.2012.163703

In this paper, the parabolic partial differential equations in three-dimensions are solved by two types of finite differences, such as, Alternating Direction Explicit (ADE) method and Alternating Direction Implicit (ADI) method. By the comparison of the numerical results for the previous two methods with the Exact solution, we observe that the results of Alternating Direction Implicit (ADI) method is better and nearest to the exact solution compared with the results of Alternating Direction Explicit (ADE) method. we also studied the numerical stability of both methods by Von-Neumann Method.

Design and Implementation of Probabilistic Assignment Management System for a Library Database Using Object Oriented Programming

Jamal M. Hameed

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 99-108
DOI: 10.33899/csmj.2012.163704

Computerized information system has recently become widely available. Indeed, it has become an important instrument that can be applied in different aspects of life. One of the important applications of computer systems is that adopted by commercial organization as well as by small business. Hence, we can say that no management can successfully operate in any business without the use of computerized information system.
This paper aims to design and implement probabilistic assignmentmanagement system for a librarydatabase commercial library. Most applications of OOP use specialized object oriented languages, but this paper shows that C++ can be used effectively to develop object oriented models of typical library management, and implement OOP is more flexible and reliable and less costing.

Application of Immune Complement Algorithm to NSL-KDD Intrusion Detection Dataset

Mafaz Mohsin Khalil Alanezi; Najlaa B. Aldabagh

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 109-123
DOI: 10.33899/csmj.2012.163705

Many real world problems involve the simultaneous optimization of various and often conflicting objectives. Evolutionary algorithms seem to be the most attractive approaches for this class of problems, because they are usually population based techniques that can find multiple compromise solution in a single run, and they do not require any hypotheses on the objective functions. Among other techniques, in the last decade a new paradigm based on the emulation of the immune system behavior has been proposed. Since the pioneer works, many different implementations have been proposed in literatures.
This Paper presents a description of an intrusion detection approach modeled on the basis of three bio-inspired concepts namely, Negative selection, Positive selection and complement system. The Positive selection mechanism of the immune system can detect the attack patterns (nonself), while the Negative selection mechanism of the immune system can delete the Artificial lymphocyte (ALC) which interact with normal patterns (Self). The complement system is a kind of the effecter mechanism, which refers to a series of proteins circulating in the blood and bathing the fluids surrounding tissues. It establishes the idea that only those cells that recognize the antigens are selected to undergo two operators: cleave operator and bind operator are presented, cleave operator cleaves a complement cell into two sub-cells, while bind operator bindtwo cells together and forms a big cell. To obtain Complement detectors can recognize only the attack patterns from the NSL-KDD dataset.

Clustering and Detecting Network Intrusion Based on Fuzzy Algorithms

Manar Y. Kashmola; Bayda I. Khaleel

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 125-138
DOI: 10.33899/csmj.2012.163706

 Clustering or (cluster analysis ) has been widely used in data analysis and pattern recognition. There are several algorithms for clustering large data sets or streaming data sets, Their aims to organize a collection of data items into clusters. These such items are more similar to each other within cluster, and difference than they are in the other clusters. Three fuzzy clustering algorithms (Fuzzy C-Means, Possibilistic C-Means and Gustafson-Kessel algorithms) were applied using kdd cup 99 data set to classify this data set  into 23 classes according to the subtype of attacks. The same data set were classified into 5 classes according to the type of attacks. In order to evaluate the performance of the system, we compute the classification rate, detection rate and false alarm rate on this data set. Finally, the results obtained from the experiments with classification rate 100% which has not been obtained in any  previous work.

The n-Hosoya Polynomial of 𝑊𝛼⊠ Cβ

Ahmed M. Ali; Haveen G. Ahmed

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 139-150
DOI: 10.33899/csmj.2012.163707

For a wheel  and a cycle  the composite graphs ⊠  is constructed from the union of  and  and  adding the edges  and  , where  is an edge of  and is an edge of . The n – diameter , the  n – Hosoya polynomial and the n – Wiener index of  ⊠  are obtained in this paper.

Construct a Simulator of a Proposed Trust and Reputation Model in (PD) P2P File Sharing Systems

Dujan B. Taha; Abdullah M. Salih

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 151-165
DOI: 10.33899/csmj.2012.163708

Trust concept is an important requirement for sustained interactions between peers, and to deal with malicious peers in P2P file sharing systems. Traditional security mechanisms and services are unable to protect against malicious behaviors, therefore trust and reputation management is considered an appropriate solution that can provide a protection against such threat. In this paper, we focused on the advantage of some existing trust models to formulate a new model that solves problems raised in the previous models. We also considered the partially decentralized (PD) peer-2-peer (P2P) architecture to execute the proposed model. Finally, we construct a C# based simulator to test proposed model on the partially decentralized P2P file sharing network. Simulation results show that the model is able to identify malicious peers effectively and isolate them from the system (sharing files), hence reducing the amount of inauthentic uploads and increasing peers’ satisfaction.

Extraction and Recognition of Color Feature in true Color Images Using Neural Network Based on Colored Histogram Technique

Orjuwan M.A. Aljawadi

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 167-181
DOI: 10.33899/csmj.2012.163709

In this research, a neural network using backpropagation (BPNN) algorithm was trained and learned to work as the cone cells in human eyes to recognize the three fundamental cells’ colors and hues, as the neural network showed good results in training and testing the color feature  it was trained and learned again to recognize two nature scenes images ; Red sunset and Blue sky images where both scenes images contain color interaction and different hues such as red-orange and blue-violet. The recognition process was based on color histogram technique in colored images which is a representation of the distribution of colors in an image by counting the number of pixels that have colors in each of a fixed list of color ranges, that span the image's color space , all possible colors in the image. The importance of this research is based on developing the ability of (BPNN) in images ‘objects recognition based on color feature that is very important feature in artificial intelligence and colored image processing fields from developing the systems of alarms robots in fire recognition , medical digenesis of tumors, certain pattern’s recognition in different segments of an image , face and eyes’ iris recognition as a part of security systems , it helps solve the problem of limitation of recognition process in neural networks in many fields.

A Hybrid Algorithm from Cuckoo Search Method with N-Queens Problem

Isra N. Alkallak

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 183-192
DOI: 10.33899/csmj.2012.163710

This paper tackled cuckoo search method. The method is one of optimization methods with N-queens problem through proposed hybridization algorithm which mimics the behavior of cuckoo to reach the solution of N-queens problem. The proposed hybridization algorithm is simulated by using MATLAB 6.5 language to mimic the behavior of some cuckoo. These laid their eggs in the nests of other host birds. If a host bird discovers that the eggs aren't their own, it will either throw these alien eggs away, or abandon its nest and build a new one. To reach the solution for N-queens problem easily. The experimental results in this paper show the effectiveness and easy implementation to reach solution.

Construct a Tool for Aspect-Oriented Analysis and Design Based on Theme Approach

Nada N. Saleem; Mohammad H. Abdulwahab

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 193-208
DOI: 10.33899/csmj.2012.163711

In this research, a (RADC-AO) tool was constructed and implemented for the requirements analysis, design and stub-code generation according to aspect-oriented (AO) concepts based on theme approach. RADC-AO automatically identifies crosscutting concerns in natural language requirements text by using natural language processing (NLP), analyze requirements and apply a set of operations on themes got in the analysis process, design classes and aspects, draw class diagram, and generates stub-code.
RADC-AO tested by input complete informal text requirements for payroll system (that contains security, logging, authorization, in addition to its core functionality which includes employees information entering, loans information entering, and payment calculation), RADC-AO successes in the test and gives good results.

On i α- Open Sets

Amir A. Mohammed; Omar y. kahtab

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 219-228
DOI: 10.33899/csmj.2012.163713

In this paper, we introduce a new class of open sets defined as follows: A subset A of a topological space  is called iα-open set, if  there exists a non-empty subset O of X, , such that. Also, we present the notion of iα-continuous mapping, iα-open mapping, iα-irresolute mapping, iα-totally continuous mapping, i-contra-continuous mapping, iα-contra-continuous mapping and we investigate some properties of these mappings. Furthermore, we introduce some iα-separation axioms and the mappings are related with iα-separation axioms.

Generating a New Hyperchaotic Pan System Via State Feedback Control

Saad Fawzi AL-Azzawi

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 229-241
DOI: 10.33899/csmj.2012.163714

This paper proposes a new four-dimensional continuous autonomous hyperchaotic system based on the 3D Pan system by introducing a nonlinear state feedback controller. Dynamical behaviors of the new system are analyzed, both theoretically and numerically, including equilibrium points, Lyapunov exponents spectrum  stability and bifurcation , finally, an illustrative example is given. 

Numerical Solution of a Reaction-Diffusion System with Fast Reversible Reaction by Using Adomian’s Decomposition Method and He’s Variational Iteration Method

Ann Jalal; Mohammed O. Al-Amr

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 243-257
DOI: 10.33899/csmj.2012.163715

In this paper, the approximate solution of a reaction-diffusion system with fast reversible reaction is obtained by using Adomian decomposition method (ADM) and variational iteration method (VIM) which are two powerful methods that were recently developed. The VIM requires the evaluation of the Lagrange multiplier, whereas ADM requires the evaluation of the Adomian polynomials. The behavior of the approximate solutions and the effects of different values of  are shown graphically.

Stability Study of Double and Spherical Pendulum

Thair Y. Thanoon; Noor Husain Abdullah

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 13-26
DOI: 10.33899/csmj.2012.163716

This paper devoted to study the stability of periodic motion for physical application which is leads to differential equations of second order )Double and Spherical Pendulum( respectively by using the stability of equilibrium position given by Laypunov and Ghetagev's methods which depends on principle of energy conservation, also we will describe periodic motion and explain the phase plane )The trajectory of solutions( and state of the stability for double and spherical pendulum by using )Maple(.

Fractal Compression of Colored Animated Pictures

Basil Thanoon Younis; Abeer Abdul Khaliq Thanoon

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 27-41
DOI: 10.33899/csmj.2012.163717

This research deals with the question of pressing two series of video images by using the pressing Fractal through the use of iterative function systems, where the film frame video acquired is divided after  each converting it to one type of the storage formulas for the image files independently. Then, the image is divided into non-overlapping range blocks with the size P×P, the biggest block of which is the overlapping domain one with the size 2P×2P.
 After that matching is made between rang blocks and each block of domain blocks to obtain domain blocks identical to range blocks. The purpose of making matching is to obtain the best domain block identical to rang block by using familiar transfers, to reduce the  volume of storage with the possibility of decompression and making a comparison between the time of compression and decompression.
 The research sample includes two series of video films from the city of Mosul, namely, the Nineveh bridge and the building of college of Computer Science and Mathematics at the University of Mosul, It is found that algorithm used is efficient as far as compression and decompression of color and moving images are concerned.

Evaluate and comparission between Software Engineering Methodologies of Agent Systems

Asmaa Yaseen Hammo; Maher Talal Alasaady

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 43-61
DOI: 10.33899/csmj.2012.163718

Agent based systems software engineering is a new model in the field of software engineering. It provides new approaches for developing agent based systems. Numerous methodologies for agent based systems software engineering have been proposed. However, it is difficult to select a methodology for a specific project, and their application is still limited due to their lack of maturity and weakness for some of them. Evaluating and comparing their strengths and weaknesses is an important step towards developing better methodologies in the future, and help developers to select the most suitable methodology for project development. This research presents an evaluation and comparison of five agent based systems methodologies: Tropos, Gaia, Multi-agent Systems Engineering (MaSE), Multi Agent Systems Development (MASD) and Smart Multi Agent System (SmartMAS) methodology that is suggested by authors. The requirements of  an agent based E-Commerce system were studied and applied on the development phases for these five methodologies, and toward a formal study, the evaluation and comparison were performed based upon a framework which addresses four major criteria: concepts and properties, notations and modeling, development process, and pragmatics. The results shows that the proposed methodology is give (85%) percentage of covering these criteria for the systems that were developed by using this methodology; this concluded that the proposed methodology is better than other methodologies.

Determined the Edges Using the ant Colony Algorithm and Apply them to Medical Images

Maha Abdul Rahman Hasso; Aseel Ismail Ali

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 63-79
DOI: 10.33899/csmj.2012.163719

Ant Colony Optimization (ACO) is a method of heuristic search using in general artificial intelligence (swarm intelligence) to simulate the behavior of the aggregate food for ants to find new solutions to the combinatorial optimization problems. Artificial ant's behavior depends on the trails of real ant with additional capabilities to make it more effective such as a memory to save the past events. Every ant build solutions to the problem, and uses the information grouped about the features and performance of the private problem, to change the look to the ant problem.
In this work, an edge detection technique based on Ant Colony Optimization is used by selecting pheromone matrix which represents the information about edges in each pixel based on the guidelines set up by the ant on the image. Multiple values for different sizes of neighbor pixels are applied and a heuristic information function to test results is proposed. The results show high accuracy in edge detection of different biomedical images with different neighbors, the proposed algorithm is implemented in C Sharp 2008 language which provides high-efficiency software visible language and speed. A comparative study is also given illustrating the superiority of the proposed algorithm.

Detection of network anomaly based on hybrid intelligence techniques

Shahbaa I. Khaleel; Karam mohammed mahdi saleh

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 81-98
DOI: 10.33899/csmj.2012.163720

Artificial Intelligence could make the use of Intrusion Detection Systems a lot easier than it is today. As always, the hardest thing with learning Artificial Intelligence systems is to make them learn the right things. This research focuses on finding out how to make an Intrusion Detection Systems environment learn the preferences and work practices of a security officer, In this research hybrid intelligence system is designed and developed for network intrusion detection, where the research was presented four methods for network anomaly detection using clustering technology and dependence on artificial intelligence techniques, which include a Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) to develop and improve the performance of intrusion detection system. The first method implemented by applying traditional clustering algorithm of KM in a way Kmeans on KDDcup99 data to detect attacks, in the way the second hybrid clustering algorithm HCA method was used where the Kmeans been hybridized with GA. In the third method PSO has been used. Depending on the third method the fourth method Modified PSO (MPSO) has been developed, This was the best method among the four methods used in this research.

DNA Sequences as a Medium for Steganography Method of Codon Frequency with Character Frequency

Sadoon H. Abdullah; Ahmed S. Nori

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 99-112
DOI: 10.33899/csmj.2012.163721

The present study included an application of new method of steganography using DNA sequence as a medium for hiding. This method is considered as a secret cover for the secret message (text, image), avoid attention of unauthorized person in addition to be inextricable since it needs high effort, long time and well background in biology.
A Sequence of DNA has been synthesized chemically depending on the text secret message via using tables of symbols formed from English letter, numbers and special characters. Each symbol has been represented on codon and the symbols were arranged in the table in descending order depending on English letter frequency and codon frequency (Genes).
The DNA sequence has been synthesized by Korean Bioneer Company (sender), then the sequence has been read in university of fatih (the receiver) by DNA Sequence 3310, the results revealed high similarity between sequence of the sender and receiver. And proved that the method is protected very well against analysis and stegoanalysis.

Traffic sign recognition by using chain code

Sundus Khaleel Ebraheem; Reham Jassim Al-Atiwi

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 113-126
DOI: 10.33899/csmj.2012.163722

The Traffic signs are considered as Traffic Safety tools, Because of their role in the organization of traffic and vehicles to insure the safety of the passengers, pedestrians and the structures of traffic signs are very important devices which help the drivers to drive in safety and adequate manner.
In this research Traffic sign detection  performed in two stages: The first stage include the traffic sign detection and extraction from the road image scene, depending on the color features of the sign, the red color  of the image was taken by using RGB color space system and applying threshold method, in which, for each layer specific threshold was applied. Considering the information of the external shape to recognize the shape geometry type of external frame by using Chain Code. While the second stage include traffic sign classification depending on inner contents of the sign, depending on the number of objects found in the inside part of the sign. Then, the Chain Code was used to recognize the boundary of the inner content of the sign. The research applied on a group of images with (.bmp, .jpg) extensions and with various sizes. The distinction percentage was (99%), the database included 30 images; 15 of them are warning and the other 15 are regulatory.

Using Genetic Algorithm to Reduce the Noise Effect on Images

Baydaa S Bhnam

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 127-142
DOI: 10.33899/csmj.2012.163723

This paper deals with a problem that concentrates on the noise removal that the images are affected from different resources employing Genetic Algorithm with filters. To achieve the aims of the paper, six types of genetic filters are suggested for noise removal from the images. These suggested genetic filters depending on filters (mean, median, min and max) as an objective function for them.
These suggested genetic filters are applied on several real images contaminated by two types of noise with different levels for comparison and to show the effectiveness of them. The result show that The fifth genetic filter that depends on the median filter as an objective function and heuristic crossover and  adding and subtracting mutation, gives the best results with RMSE=15.7243 and PSNR=24.1646 for Lena.bmp image and with RMSE=8.6197 and PSNR=29.4210 for girl.png image when add 0.05 salt & paper noise.

Sound Signal De-noising Using Wavelet Transform

Yusra Faisal Al-Irhaim; Ahmad Mohammd Suliman

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 143-157
DOI: 10.33899/csmj.2012.163724

In this paper, the Discrete Wavelet Transform is studied in purifying the sound signal from noise because of the good capabilities in this scope, especially when it is merged with both types of the (Thresholding), the solid and the flexible. The aim of this research is making comparisons between the types of deferent  Discrete Wavelet Transform for both the filters which are used (Harr, Daubechies) in deferent levels (2, 3, 4, 5) with the additive of two types of noise to these filters (Gaussian White Noise) and (Random Noise). The decibel value that added to these filters was in the (5dB,10dB,15dB) values. Good results of the purification process are achived after computing the (Signal to Noise Ratio (SNR)) and (Mean Sequre Error (MSE)).

Compression of Digital Image based on Hybrid Heuristic Algorithm

Fawziya Mahmood Ramo; Yaser Noor Al Deen

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 159-171
DOI: 10.33899/csmj.2012.163725

In this research paper a system has been proposed to be used in data compression of digital image  based on two hybrid intelligent Algorithms. In the first algorithm which is now as  Meta Heuristic Genetic Compression Algorithm (MGCA) the characteristic and features of GA and local search are used to compress digital image. The second algorithm is the (HMGTCA) Hybrid Meta Genetic and Tabu Compression Algorithm. Hybrid operation has been done between Meta Heuristic Genetic and Tabu search algorithm. The proposed algorithm has been applied on four  samples. Efficiencies measures has been performed pled to calculate the value of (PSNR, MSE, correlation coefficient, compression ration and calculate the performance time). The experiments showed that the proposed algorithm achieved high performance and produces PSNR=  34.

w-Hosoya Polynomials for Connection for Some Special Graphs

Ahmed M. Ali; Assma Salah Aziz; Hafen Jalal Ahmed

AL-Rafidain Journal of Computer Sciences and Mathematics, Volume 9, Issue 2, Pages 173-184
DOI: 10.33899/csmj.2012.163726

Let  and  be any two distinct vertices in a connected graph . A container is a set of internally disjoint - paths. The width of is denoted by  or  is , and the length of  is the length of the longest - path in . Then, for a given positive integer w, the width distance between any two distinct vertices u and v in a connected graph  is define by: , where the minimum is taken over all containers of width w.
In  this paper, we find the Hosoya polynomials, and Wiener indices of the join of two special graphs such as bipartite complete graphs, paths, cycles, star graphs and wheel graphs with respect  to the width distance.