This document discusses zero-divisor graphs for direct products of finite commutative rings. Specifically, it analyzes the zero-divisor graphs, neighborhoods, and adjacency matrices for the rings Zp × Zp2, Zp × Z2p, and Zp × Zp2–2, where p is a prime number. It presents the constructions of the zero-divisor graphs for examples when p=2 and p=3. It also generalizes the results to the ring Zp × Zp2 for any prime p. The document proves properties of the adjacency matrices and neighborhoods for these zero-divisor graphs. Finally, it discusses some results on annihilators for the zero-divisor graph of the direct product of any two
Mathematics and History of Complex VariablesSolo Hermelin
Mathematics of complex variables, plus history.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com, thanks! For more presentations, please visit my website at http://www.solohermelin.com
Introduction of metric dimension of circular graphs is connected graph , The distance and diameter , Resolving sets and location number then Examples . Application in facility location problems . is has motivation (Applications in Chemistry and Networks systems). Definitions of Certain Regular Graphs. Main Results for three graphs (Prism , Antiprism and generalized Petersen graphs .
Mathematics and History of Complex VariablesSolo Hermelin
Mathematics of complex variables, plus history.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com, thanks! For more presentations, please visit my website at http://www.solohermelin.com
Introduction of metric dimension of circular graphs is connected graph , The distance and diameter , Resolving sets and location number then Examples . Application in facility location problems . is has motivation (Applications in Chemistry and Networks systems). Definitions of Certain Regular Graphs. Main Results for three graphs (Prism , Antiprism and generalized Petersen graphs .
Aplikasi Bilangan Kompleks - Analisis Sinyal [PAPER]Ghins GO
Check out!
Website : https://ghinsblog.blogspot.com
Youtube : Ghins GO Math
APLIKASI BILANGAN KOMPLEKS
BAB 1 Pendahuluan
1. Latar belakang
2. Rumusan Masalah
3. Tujuan
BAB 2 Pembahasan
1. Bilangan Kompleks
2. Sinyal
BAB 3 Aplikasi
1. Analisis Sinyal
2. Analisis Fourier
3. Analisis Frekuensi Sinyal Periodik Waktu Diskrit
4. Penerapan Analisis Sinyal
BAB 4 Kesimpulan
DAFTAR PUSTAKA
[1] Gumilang, Muhammad. 2015. "Aplikasi Bilangan Kompleks dalam Analisis Sinyal". https://adoc.pub/aplikasi-bilangan-kompleks-dalam-analisis-sinyal.html, diakses 14 Mei 2021 pukul 18.11.
[2] Haryono, Stefanus Agus. 2015."Penggunaan Bilangan Kompleks dalam Pemrosesan Signal".https://pdfslide.tips/documents/penggunaan-bilangan-kompleks-dalam-pemrosesan-signalinformatikasteiitbacidrinaldimuniraljabargeometri2015aa.html, diakses 13 Mei 2021 pukul 20.37.
[5] Ratnadewi. dkk. 2019. "MATEMATIKA TEKNIK". Bandung: Rekayasa Sains.
Selengkapnya:
https://ghinsblog.blogspot.com/2021/08/variabel-kompleks-aplikasi-bilangan.html
In this chapter, the authors extend the theory of
the generalized difference Operator ∆L to the generalized
difference operator of the 풏
풕풉kind denoted by ∆L Where L
=푳 = {풍ퟏ,풍ퟐ,….풍풏} of positive reals풍ퟏ,풍ퟐ,….풍풏and obtain some
interesting results on the relation between the generalized
polynomial factorial of the first kind, 풏
풕풉kind and algebraic
polynomials. Also formulae for the sum of the general
partial sums of products of several powers of consecutive
terms of an Arithmetic progression in number theory are
derived.
A New Approach to Segmentation of On-Line Persian Cursive WordsEditor IJCATR
Segmentation approaches, as processes that divide word into smaller parts which contain one letter at most, have important
effect on cursive word recognition. While online cursive word recognition became applied technology in Latin and Chinese languages,
complex structural features in Arabic-based script made it an important field of study in Persian and Arabic languages. In this paper,
by introducing of Standard Persian Handwriting, we proposed a novel approach to segmentation online Persian cursive script based on
width of letter's body in Persian language. Results are shown 99.86% accuracy in detection of expected segmentation points, while
recognized extra points reduced 93.73% compared to our previous methods.
Ijcatr03051008Implementation of Matrix based Mapping Method Using Elliptic Cu...Editor IJCATR
Elliptic Curve Cryptography (ECC) gained a lot of attention in industry. The key attraction of ECC over RSA is that it
offers equal security even for smaller bit size, thus reducing the processing complexity. ECC Encryption and Decryption methods can
only perform encrypt and decrypt operations on the curve but not on the message. This paper presents a fast mapping method based on
matrix approach for ECC, which offers high security for the encrypted message. First, the alphabetic message is mapped on to the
points on an elliptic curve. Later encode those points using Elgamal encryption method with the use of a non-singular matrix. And the
encoded message can be decrypted by Elgamal decryption technique and to get back the original message, the matrix obtained from
decoding is multiplied with the inverse of non-singular matrix. The coding is done using Verilog. The design is simulated and
synthesized using FPGA.
Aplikasi Bilangan Kompleks - Analisis Sinyal [PAPER]Ghins GO
Check out!
Website : https://ghinsblog.blogspot.com
Youtube : Ghins GO Math
APLIKASI BILANGAN KOMPLEKS
BAB 1 Pendahuluan
1. Latar belakang
2. Rumusan Masalah
3. Tujuan
BAB 2 Pembahasan
1. Bilangan Kompleks
2. Sinyal
BAB 3 Aplikasi
1. Analisis Sinyal
2. Analisis Fourier
3. Analisis Frekuensi Sinyal Periodik Waktu Diskrit
4. Penerapan Analisis Sinyal
BAB 4 Kesimpulan
DAFTAR PUSTAKA
[1] Gumilang, Muhammad. 2015. "Aplikasi Bilangan Kompleks dalam Analisis Sinyal". https://adoc.pub/aplikasi-bilangan-kompleks-dalam-analisis-sinyal.html, diakses 14 Mei 2021 pukul 18.11.
[2] Haryono, Stefanus Agus. 2015."Penggunaan Bilangan Kompleks dalam Pemrosesan Signal".https://pdfslide.tips/documents/penggunaan-bilangan-kompleks-dalam-pemrosesan-signalinformatikasteiitbacidrinaldimuniraljabargeometri2015aa.html, diakses 13 Mei 2021 pukul 20.37.
[5] Ratnadewi. dkk. 2019. "MATEMATIKA TEKNIK". Bandung: Rekayasa Sains.
Selengkapnya:
https://ghinsblog.blogspot.com/2021/08/variabel-kompleks-aplikasi-bilangan.html
In this chapter, the authors extend the theory of
the generalized difference Operator ∆L to the generalized
difference operator of the 풏
풕풉kind denoted by ∆L Where L
=푳 = {풍ퟏ,풍ퟐ,….풍풏} of positive reals풍ퟏ,풍ퟐ,….풍풏and obtain some
interesting results on the relation between the generalized
polynomial factorial of the first kind, 풏
풕풉kind and algebraic
polynomials. Also formulae for the sum of the general
partial sums of products of several powers of consecutive
terms of an Arithmetic progression in number theory are
derived.
A New Approach to Segmentation of On-Line Persian Cursive WordsEditor IJCATR
Segmentation approaches, as processes that divide word into smaller parts which contain one letter at most, have important
effect on cursive word recognition. While online cursive word recognition became applied technology in Latin and Chinese languages,
complex structural features in Arabic-based script made it an important field of study in Persian and Arabic languages. In this paper,
by introducing of Standard Persian Handwriting, we proposed a novel approach to segmentation online Persian cursive script based on
width of letter's body in Persian language. Results are shown 99.86% accuracy in detection of expected segmentation points, while
recognized extra points reduced 93.73% compared to our previous methods.
Ijcatr03051008Implementation of Matrix based Mapping Method Using Elliptic Cu...Editor IJCATR
Elliptic Curve Cryptography (ECC) gained a lot of attention in industry. The key attraction of ECC over RSA is that it
offers equal security even for smaller bit size, thus reducing the processing complexity. ECC Encryption and Decryption methods can
only perform encrypt and decrypt operations on the curve but not on the message. This paper presents a fast mapping method based on
matrix approach for ECC, which offers high security for the encrypted message. First, the alphabetic message is mapped on to the
points on an elliptic curve. Later encode those points using Elgamal encryption method with the use of a non-singular matrix. And the
encoded message can be decrypted by Elgamal decryption technique and to get back the original message, the matrix obtained from
decoding is multiplied with the inverse of non-singular matrix. The coding is done using Verilog. The design is simulated and
synthesized using FPGA.
Successive approximation of neutral stochastic functional differential equati...Editor IJCATR
We establish results concerning the existence and uniqueness of solutions to neutral stochastic functional differential
equations with infinite delay and Poisson jumps in the phase space C((-∞,0];Rd) under non-Lipschitz condition with Lipschitz
condition being considered as a special case and a weakened linear growth condition on the coefficients by means of the successive
approximation. Compared with the previous results, the results obtained in this paper is based on a other proof and our results can
complement the earlier publications in the existing literatures.
Modeling and Evaluation of Performance and Reliability of Component-based So...Editor IJCATR
Validation of software systems is very useful at the primary stages of their development cycle. Evaluation of functional
requirements is supported by clear and appropriate approaches, but there is no similar strategy for evaluation of non-functional requirements
(such as performance and reliability). Whereas establishing the non-functional requirements have significant effect on success of software
systems, therefore considerable necessities are needed for evaluation of non-functional requirements. Also, if the software performance has
been specified based on performance models, may be evaluated at the primary stages of software development cycle. Therefore, modeling
and evaluation of non-functional requirements in software architecture level, that are designed at the primary stages of software systems
development cycle and prior to implementation, will be very effective.
We propose an approach for evaluate the performance and reliability of software systems, based on formal models (hierarchical timed
colored petri nets) in software architecture level. In this approach, the software architecture is described by UML use case, activity and
component diagrams, then UML model is transformed to an executable model based on hierarchical timed colored petri nets (HTCPN) by a
proposed algorithm. Consequently, upon execution of an executive model and analysis of its results, non-functional requirements including
performance (such as response time) and reliability may be evaluated in software architecture level.
Second order and Third order NLO studies of L- alanine crystals grown in aque...Editor IJCATR
Nonlinear optics is a fascinating field, which plays a vital role in the emerging field of photonics and optoelectronics. A
nonlinear optical crystal of L-alanine grown in aqueous solution of hydrofluoric acid is done by slow evaporation method. L-alanine is
an NLO material and it has a Second Harmonic Generation (SHG) efficiency of about 0.3 times that of KDP. To alter the various
properties of L-alanine, single crystals of L-alanine have been grown in the aqueous solution of hydrofluoric acid. In this work, Lalanine
was admixtured with hydrofluoric acid (LAHF) in the molar ratio of 1:1. The grown crystals were colorless and transparent
and they were subjected to various studies for characterization.The third-order nonlinearities of LAHF crystal have been investigated
by Z-scan method. The values of nonlinear refractive index (n2), the nonlinear absorption coefficient (β) and third-order nonlinear
susceptibility (χ(3)) are estimated for the sample
RW-CLOSED MAPS AND RW-OPEN MAPS IN TOPOLOGICAL SPACESEditor IJCATR
In this paper we introduce rw-closed map from a topological space X to a topological space Y as the image
of every closed set is rw-closed and also we prove that the composition of two rw-closed maps need not be rw-closed
map. We also obtain some properties of rw-closed maps.
Formal Models for Context Aware ComputingEditor IJCATR
Context-aware computing refers to a general class of mobile systems that can sense their physical environment, and
adapt their behavior accordingly. In this paper we seek to develop a systematic understanding of context-aware computing by
constructing a formal model and notation for expressing context-aware computations. This discussion is followed by a
description and comparison of current context modeling and reasoning techniques.
Natural Hand Gestures Recognition System for Intelligent HCI: A SurveyEditor IJCATR
Gesture recognition is to recognizing meaningful expressions of motion by a human, involving the hands, arms, face, head,
and/or body. Hand Gestures have greater importance in designing an intelligent and efficient human–computer interface. The applications
of gesture recognition are manifold, ranging from sign language through medical rehabilitation to virtual reality. In this paper a survey on
various recent gesture recognition approaches is provided with particular emphasis on hand gestures. A review of static hand posture
methods are explained with different tools and algorithms applied on gesture recognition system, including connectionist models, hidden
Markov model, and fuzzy clustering. Challenges and future research directions are also highlighted.
A Study of Sybil and Temporal Attacks in Vehicular Ad Hoc Networks: Types, Ch...Editor IJCATR
In recent years, the number of automobiles on the road has increased tremendously. Due to high density and mobility of vehicles,
possible threats and road accidents are increasing. Wireless communication allows sending safety and other critical information. Due to this
inherent wireless characteristic and periodic exchange of safety packets, Vehicular Ad-hoc Network (VANET) is vulnerable to number of
security threats like Sybil attack or temporal attack. In this paper, a detailed discussion has been done on both the type of attacks. With the
help of already published works, some approaches have also been studied which have proved to be of significance in detection of these
attacks.
Survey on Efficient and Secure Anonymous Communication in ManetsEditor IJCATR
Mobile ad-hoc networks require anonymous communications in order to thwart new wireless passive attacks; and to protect new
assets of information such as nodes locations, motion patterns, network topology and traffic patterns in addition to conventional identity and
message privacy. The transmitted routing messages and cached active routing entries leave plenty of opportunities for eavesdroppers.
Anonymity and location privacy guarantees for the deployed ad hoc networks are critical in military and real time communication systems,
otherwise the entire mission may be compromised. This poses challenging constraints on MANET routing and data forwarding. To address
the new challenges, several anonymous routing schemes have been proposed recently.
Optical Character Recognition from Text ImageEditor IJCATR
Optical Character Recognition (OCR) is a system that provides a full alphanumeric recognition of printed or handwritten
characters by simply scanning the text image. OCR system interprets the printed or handwritten characters image and converts it into
corresponding editable text document. The text image is divided into regions by isolating each line, then individual characters with
spaces. After character extraction, the texture and topological features like corner points, features of different regions, ratio of
character area and convex area of all characters of text image are calculated. Previously features of each uppercase and lowercase
letter, digit, and symbols are stored as a template. Based on the texture and topological features, the system recognizes the exact
character using feature matching between the extracted character and the template of all characters as a measure of similarity.
Randić Index of Some Class of Trees with an AlgorithmEditor IJCATR
The Randić index R(G) of a graph G is defined as the sum of the weights (dG(u)dG(v))-1/2 over all edges e = uv of G. In this
paper we have obtained the Randić index of some class of trees and of their complements. Also further developed an algorithmic
technique to find Randić index of a graph.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
Toeplitz Hermitian Positive Definite (THPD) matrices play an important role in signal processing and
computer graphics and circular models, related to angular / periodic data, have vide applications in
various walks of life. Visualizing a circular model through THPD matrix the required computations on
THPD matrices using single bordering and double bordering are discussed. It can be seen that every
tridiagonal THPD leads to Cardioid model.
Toeplitz Hermitian Positive Definite (THPD) matrices play an important role in signal processing and computer graphics and circular models, related to angular / periodic data, have vide applications in various walks of life. Visualizing a circular model through THPD matrix the required computations on THPD matrices using single bordering and double bordering are discussed. It can be seen that every tridiagonal THPD leads to Cardioid model.
Representation of Integer Positive Number as A Sum of Natural SummandsIJERA Editor
In this paper the problem of representation of integer positive number as a sum of natural terms is considered. The new approach to calculation of number of representations is offered. Results of calculations for numbers from 1 to 500 are given. Dependence of partial contributions to total sum of number of representations is investigated. Application of results is discussed.
On algorithmic problems concerning graphs of higher degree of symmetrygraphhoc
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry. The complexity of computing the adjacency matrices of a graph Gr on the vertices X such that
Aut GR = G depends very much on the description of the geometry with which one starts. For example, we
can represent the geometry as the totality of 1 cosets of parabolic subgroups 2 chains of embedded
subspaces (case of linear groups), or totally isotropic subspaces (case of the remaining classical groups), 3
special subspaces of minimal module for G which are defined in terms of a G invariant multilinear form.
The aim of this research is to develop an effective method for generation of graphs connected with classical
geometry and evaluation of its spectra, which is the set of eigenvalues of adjacency matrix of a graph. The
main approach is to avoid manual drawing and to calculate graph layout automatically according to its
formal structure. This is a simple task in a case of a tree like graph with a strict hierarchy of entities but it
becomes more complicated for graphs of geometrical nature. There are two main reasons for the
investigations of spectra: (1) very often spectra carry much more useful information about the graph than a
corresponding list of entities and relationships (2) graphs with special spectra, satisfying so called
Ramanujan property or simply Ramanujan graphs (by name of Indian genius mathematician) are important
for real life applications (see [13]). There is a motivated suspicion that among geometrical graphs one
could find some new Ramanujan graphs.
ON ALGORITHMIC PROBLEMS CONCERNING GRAPHS OF HIGHER DEGREE OF SYMMETRYFransiskeran
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Research Inventy : International Journal of Engineering and Scienceinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
International Journal of Engineering Research and Development is an international premier peer reviewed open access engineering and technology journal promoting the discovery, innovation, advancement and dissemination of basic and transitional knowledge in engineering, technology and related disciplines.
In this presentation, a more accurate expression of the zeta zero-counting function is developed exhibiting the expected step function behavior and its relation to the primes is demonstrated.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
Text Mining in Digital Libraries using OKAPI BM25 ModelEditor IJCATR
The emergence of the internet has made vast amounts of information available and easily accessible online. As a result, most libraries have digitized their content in order to remain relevant to their users and to keep pace with the advancement of the internet. However, these digital libraries have been criticized for using inefficient information retrieval models that do not perform relevance ranking to the retrieved results. This paper proposed the use of OKAPI BM25 model in text mining so as means of improving relevance ranking of digital libraries. Okapi BM25 model was selected because it is a probability-based relevance ranking algorithm. A case study research was conducted and the model design was based on information retrieval processes. The performance of Boolean, vector space, and Okapi BM25 models was compared for data retrieval. Relevant ranked documents were retrieved and displayed at the OPAC framework search page. The results revealed that Okapi BM 25 outperformed Boolean model and Vector Space model. Therefore, this paper proposes the use of Okapi BM25 model to reward terms according to their relative frequencies in a document so as to improve the performance of text mining in digital libraries.
Green Computing, eco trends, climate change, e-waste and eco-friendlyEditor IJCATR
This study focused on the practice of using computing resources more efficiently while maintaining or increasing overall performance. Sustainable IT services require the integration of green computing practices such as power management, virtualization, improving cooling technology, recycling, electronic waste disposal, and optimization of the IT infrastructure to meet sustainability requirements. Studies have shown that costs of power utilized by IT departments can approach 50% of the overall energy costs for an organization. While there is an expectation that green IT should lower costs and the firm’s impact on the environment, there has been far less attention directed at understanding the strategic benefits of sustainable IT services in terms of the creation of customer value, business value and societal value. This paper provides a review of the literature on sustainable IT, key areas of focus, and identifies a core set of principles to guide sustainable IT service design.
Policies for Green Computing and E-Waste in NigeriaEditor IJCATR
Computers today are an integral part of individuals’ lives all around the world, but unfortunately these devices are toxic to the environment given the materials used, their limited battery life and technological obsolescence. Individuals are concerned about the hazardous materials ever present in computers, even if the importance of various attributes differs, and that a more environment -friendly attitude can be obtained through exposure to educational materials. In this paper, we aim to delineate the problem of e-waste in Nigeria and highlight a series of measures and the advantage they herald for our country and propose a series of action steps to develop in these areas further. It is possible for Nigeria to have an immediate economic stimulus and job creation while moving quickly to abide by the requirements of climate change legislation and energy efficiency directives. The costs of implementing energy efficiency and renewable energy measures are minimal as they are not cash expenditures but rather investments paid back by future, continuous energy savings.
Performance Evaluation of VANETs for Evaluating Node Stability in Dynamic Sce...Editor IJCATR
Vehicular ad hoc networks (VANETs) are a favorable area of exploration which empowers the interconnection amid the movable vehicles and between transportable units (vehicles) and road side units (RSU). In Vehicular Ad Hoc Networks (VANETs), mobile vehicles can be organized into assemblage to promote interconnection links. The assemblage arrangement according to dimensions and geographical extend has serious influence on attribute of interaction .Vehicular ad hoc networks (VANETs) are subclass of mobile Ad-hoc network involving more complex mobility patterns. Because of mobility the topology changes very frequently. This raises a number of technical challenges including the stability of the network .There is a need for assemblage configuration leading to more stable realistic network. The paper provides investigation of various simulation scenarios in which cluster using k-means algorithm are generated and their numbers are varied to find the more stable configuration in real scenario of road.
Optimum Location of DG Units Considering Operation ConditionsEditor IJCATR
The optimal sizing and placement of Distributed Generation units (DG) are becoming very attractive to researchers these days. In this paper a two stage approach has been used for allocation and sizing of DGs in distribution system with time varying load model. The strategic placement of DGs can help in reducing energy losses and improving voltage profile. The proposed work discusses time varying loads that can be useful for selecting the location and optimizing DG operation. The method has the potential to be used for integrating the available DGs by identifying the best locations in a power system. The proposed method has been demonstrated on 9-bus test system.
Analysis of Comparison of Fuzzy Knn, C4.5 Algorithm, and Naïve Bayes Classifi...Editor IJCATR
Early detection of diabetes mellitus (DM) can prevent or inhibit complication. There are several laboratory test that must be done to detect DM. The result of this laboratory test then converted into data training. Data training used in this study generated from UCI Pima Database with 6 attributes that were used to classify positive or negative diabetes. There are various classification methods that are commonly used, and in this study three of them were compared, which were fuzzy KNN, C4.5 algorithm and Naïve Bayes Classifier (NBC) with one identical case. The objective of this study was to create software to classify DM using tested methods and compared the three methods based on accuracy, precision, and recall. The results showed that the best method was Fuzzy KNN with average and maximum accuracy reached 96% and 98%, respectively. In second place, NBC method had respective average and maximum accuracy of 87.5% and 90%. Lastly, C4.5 algorithm had average and maximum accuracy of 79.5% and 86%, respectively.
Web Scraping for Estimating new Record from Source SiteEditor IJCATR
Study in the Competitive field of Intelligent, and studies in the field of Web Scraping, have a symbiotic relationship mutualism. In the information age today, the website serves as a main source. The research focus is on how to get data from websites and how to slow down the intensity of the download. The problem that arises is the website sources are autonomous so that vulnerable changes the structure of the content at any time. The next problem is the system intrusion detection snort installed on the server to detect bot crawler. So the researchers propose the use of the methods of Mining Data Records and the method of Exponential Smoothing so that adaptive to changes in the structure of the content and do a browse or fetch automatically follow the pattern of the occurrences of the news. The results of the tests, with the threshold 0.3 for MDR and similarity threshold score 0.65 for STM, using recall and precision values produce f-measure average 92.6%. While the results of the tests of the exponential estimation smoothing using ? = 0.5 produces MAE 18.2 datarecord duplicate. It slowed down to 3.6 datarecord from 21.8 datarecord results schedule download/fetch fix in an average time of occurrence news.
Evaluating Semantic Similarity between Biomedical Concepts/Classes through S...Editor IJCATR
Most of the existing semantic similarity measures that use ontology structure as their primary source can measure semantic similarity between concepts/classes using single ontology. The ontology-based semantic similarity techniques such as structure-based semantic similarity techniques (Path Length Measure, Wu and Palmer’s Measure, and Leacock and Chodorow’s measure), information content-based similarity techniques (Resnik’s measure, Lin’s measure), and biomedical domain ontology techniques (Al-Mubaid and Nguyen’s measure (SimDist)) were evaluated relative to human experts’ ratings, and compared on sets of concepts using the ICD-10 “V1.0” terminology within the UMLS. The experimental results validate the efficiency of the SemDist technique in single ontology, and demonstrate that SemDist semantic similarity techniques, compared with the existing techniques, gives the best overall results of correlation with experts’ ratings.
Semantic Similarity Measures between Terms in the Biomedical Domain within f...Editor IJCATR
The techniques and tests are tools used to define how measure the goodness of ontology or its resources. The similarity between biomedical classes/concepts is an important task for the biomedical information extraction and knowledge discovery. However, most of the semantic similarity techniques can be adopted to be used in the biomedical domain (UMLS). Many experiments have been conducted to check the applicability of these measures. In this paper, we investigate to measure semantic similarity between two terms within single ontology or multiple ontologies in ICD-10 “V1.0” as primary source, and compare my results to human experts score by correlation coefficient.
A Strategy for Improving the Performance of Small Files in Openstack Swift Editor IJCATR
This is an effective way to improve the storage access performance of small files in Openstack Swift by adding an aggregate storage module. Because Swift will lead to too much disk operation when querying metadata, the transfer performance of plenty of small files is low. In this paper, we propose an aggregated storage strategy (ASS), and implement it in Swift. ASS comprises two parts which include merge storage and index storage. At the first stage, ASS arranges the write request queue in chronological order, and then stores objects in volumes. These volumes are large files that are stored in Swift actually. During the short encounter time, the object-to-volume mapping information is stored in Key-Value store at the second stage. The experimental results show that the ASS can effectively improve Swift's small file transfer performance.
Integrated System for Vehicle Clearance and RegistrationEditor IJCATR
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On the Adjacency Matrix and Neighborhood Associated with Zero-divisor Graph for Direct Product of Finite Commutative Rings
1. International Journal of Computer Applications Technology and Research
Volume 2– Issue 3, 315 - 323, 2013
www.ijcat.com 315
On the Adjacency Matrix and Neighborhood Associated
with Zero-divisor Graph for Direct Product of Finite
Commutative Rings
Kuntala Patra
Department of Mathematics
Gauhati University
Guwahati - 781014, India
Priyanka Pratim Baruah
Department of Mathematics
Girijananda Chowdhury Institute of Management and
Technology
Guwahati - 781017, India
Abstract: The main purpose of this paper is to study the zero-divisor graph for direct product of finite commutative rings. In our
present investigation we discuss the zero-divisor graphs for the following direct products: direct product of the ring of integers under
addition and multiplication modulo p and the ring of integers under addition and multiplication modulo p2
for a prime number p,
direct product of the ring of integers under addition and multiplication modulo p and the ring of integers under addition and
multiplication modulo 2p for an odd prime number p and direct product of the ring of integers under addition and multiplication
modulo p and the ring of integers under addition and multiplication modulo p2
– 2 for that odd prime p for which p2
– 2 is a prime
number. The aim of this paper is to give some new ideas about the neighborhood, the neighborhood number and the adjacency matrix
corresponding to zero-divisor graphs for the above mentioned direct products. Finally, we prove some results of annihilators on zero-
divisor graph for direct product of A and B for any two commutative rings A and B with unity
Keywords: Zero-divisor, Commutative ring, Adjacency matrix, Neighborhood, Zero-divisor graph, Annihilator.
AMS Classification (2010): 05Cxx; 05C25; 05C50.
1. INTRODUCTION
The idea of zero-divisor graph of a commutative
ring was first introduced by I. Beck [2] in 1988. D. F.
Anderson and P.S. Livinsgston [1] redefined the concept of
zero-divisor graph in 1999. F. R. DeMeyer, T. Mckenzie and
K. Schneider [3] extended the concept of zero-divisor graph
for commutative semi-group in 2002. The notion of zero-
divisor graph had been extended for non-commutative rings
by S. P. Redmond [9] in 2002. Recently, P. Sharma, A.
Sharma and R. K. Vats [10] have discussed the neighborhood
set, the neighborhood number and the adjacency matrix of
zero-divisor graphs for the rings pp ZZ and
],[][ iZiZ pp where p is a prime number.
In this paper R1 denotes the finite commutative ring
such that R1 = 2
pp ZZ ( p is a prime number), R2 denotes
the finite commutative ring such that R2 = pp ZZ 2 ( p is
an odd prime number) and R3 denotes the finite commutative
ring such that R3 =
22
pp ZZ ( for that odd prime p for
which p2
– 2 is a prime number). Let R be a commutative ring
with unity and Z(R) be the set of zero-divisors of R; that is
Z(R) = {xR: xy = 0 or yx = 0 for some yR* = R –
{0}}.Then zero-divisor graph of R is an undirected graph
Γ(R) with vertex set Z(R)* = Z(R) – {0} such that distinct
vertices x and y of Z(R)* are adjacent if and only if xy = 0.
The neighborhood (or open neighborhood) NG(v) of a vertex
v of a graph G is the set of vertices adjacent to v. The closed
neighborhood NG[v] of a vertex v is the set NG(v) {v}. For
a set S of vertices, the neighborhood of S is the union of the
neighborhoods of the vertices and so it is the set of all vertices
adjacent to at least one member of S. For a graph G with
vertex set V, the union of the neighborhoods of all the vertices
is neighborhood of V and it is denoted by NG(V). The
neighborhood number nG(V) is the cardinality of NG(V). If the
graph G with vertex set V is connected, then NG(V) is the
vertex set V and the cardinality of NG(V) is equal to the
cardinality of V. If Γ(R) is the zero-divisor graph of a
commutative ring R with vertex set Z(R)* and since zero-
divisor graph is always connected [1], we have NΓ(R)(Z(R)*) =
Z(R)* andNΓ(R)(Z(R)*) = Z(R)*). Throughout this paper
(G) denotes the maximum degree of a graph G and (G)
denotes the minimum degree of a graph G. The adjacency
matrix corresponding to zero-divisor graph G is defined as
A = [aij], where aij = 1, if vi vj = 0 for any vertex vi and vj of
G and aij = 0, otherwise.
In this paper, we construct zero-divisor graphs for
the rings R1, R2 and R3. We obtain the neighborhood and the
adjacency matrices corresponding to zero-divisor graphs of
R1, R2 and R3. Some properties of adjacency matrices are also
obtained. We prove some theorems related to neighborhood
and adjacency matrices corresponding to zero-divisor graphs
of R1, R2 and R3. Finally, we prove some results of annihilators
on zero-divisor graph of A B, for any two commutative rings
A and B with unity.
2. International Journal of Computer Applications Technology and Research
Volume 2– Issue 3, 315 - 323, 2013
www.ijcat.com 316
2. CONSTRUCTION OF ZERO -DIVISOR
GRAPH FOR R1 = 2
pp ZZ ( p IS A PRIME
NUMBER):
First, we construct the zero-divisor graph for the
ring R1 = 2
pp ZZ (p is a prime number) and analyze the
graph. We start with the cases p = 2 and p = 3 and then
generalize the cases.
Case1: When p = 2 we have R1 = Z2×Z4.
The ring R1 has 5 non-zero zero-divisors. In this case V =
Z(R1)* = {(1,0), (0,1), (0,2), (0,3), (1,2)} and the zero-divisor
graph G = Γ(R1) is given by:
Fig: 1
The closed neighborhoods of the vertices are
NG[(1,0)] = {(1,0), (0,1), (0,2),(0,3)}, NG[(0,1)] ={(1,0),(0,1)},
NG[(0,2)] = {(1,0), (1,2), (0,2)}, NG[(0,3)] = {(1,0),(0,3)} and
NG[(1,2)] = {(0,2),(1,2)}. The neighborhood of V is given by
NG(V) ={(1,0), (0,1), (0,2), (0,3), (1,2)}. The maximum degree
is (G) = 3 and minimum degree is (G) = 1. The
adjacency matrix for the zero-divisor graph of R1 = Z2×Z4 is
M1 =
55
31
133313
31
00
00
T
T
B
BOA
A
where, 31A = [1 1 1],
13B =
0
1
0
, 13
T
A is the transpose of 31A , 31
T
B is
the transpose of 13B and 33O is the zero matrix.
Properties of adjacency matrix M1:
(i) The determinant of the adjacency matrix M1 corresponding
to G = Γ(R1) is 0.
(ii) The rank of the adjacency matrix M1 corresponding to G =
Γ(R1) is 2.
(iii) The adjacency matrix M1 corresponding to G = Γ(R1) is
symmetric and singular.
Case2: When p = 3 we have R1 = Z3×Z9.
The ring R1 has 14 non-zero zero-divisors. In this case V =
Z(R1)* = {(1,0), (2,0), (1,3), (1,6), (2,3), (2,6), (0,1), (0,2),
(0,3), (0,4), (0,5),(0,6),(0,7),(0,8)} and the zero-divisor graph
G = Γ(R1) is given by:
Fig: 2
The closed neighborhoods of the vertices are
NG[(1,0)] = {(0,1), (0,2), (0,3), (0,4), (0,5), (0,6), (0,7), (0,8),
(1,0)}, NG[(2,0)] ={(0,1), (0,2), (0,3), (0,4), (0,5), (0,6), (0,7),
(0,8),(2,0)}, NG[(1,3)] = {(0,3),(0,6),(1,3)}, NG[(1,6)] = {(0,3),
(0,6),(1,6)}, NG[(2,3)] = {(0,3),(0,6),(2,3)}, NG[(2,6)] = {(0,3),
(0,6),(2,6)}, NG[(0,1)] = {(1,0),(2,0),(0,1)}, NG[(0,2)] = {(1,0),
(2,0),(0,2)}, NG[(0,3)] = {(1,0), (2,0), (1,3), (1,6), (2,3), (2,6),
(0,6),(0,3)}, NG[(0,4)] ={(1,0),(2,0),(0,4)}, NG[(0,5)] ={(1,0),
(2,0),(0,5)}, NG[(0,6)] = {(1,0), (2,0), (1,3), (1,6), (2,3), (2,6),
(0,3),(0,6)}, NG[(0,7)] ={(1,0), (2,0), (0,7)}, NG[(0,8)] ={(1,0),
(2,0), (0,8)}. The neighborhood of V is given by NG(V) =
{(1,0), (0,2), (1,3), (1,6), (2,3), (2,6), (0,1), (0,2), (0,3), (0,4),
(0,5), (0,6), (0,7), (0,8)}. The maximum degree is (G) = 8
and minimum degree is (G) = 2. The adjacency matrix for
the zero-divisor graph of R1 = Z3×Z9 is M1 ═
1414335363
355565
365666
OCB
COA
BAO
TT
T where A65 =
00100
00100
00100
00100
11111
11111
,
B63 =
001
001
001
001
111
111
, C53 =
000
000
001
000
000
, O66, O55, O33 are
the zero matrices and AT
56, BT
36, CT
35 are the transposes of
A65, B63, C53 respectively.
Properties of adjacency matrix M1:
3. International Journal of Computer Applications Technology and Research
Volume 2– Issue 3, 315 - 323, 2013
www.ijcat.com 317
(i) The determinant of the adjacency matrix M1 corresponding
to G = Γ(R1) is 0.
(ii) The rank of the adjacency matrix M1 corresponding to G =
Γ(R1) is 2.
(iii) The adjacency matrix M1 corresponding to G = Γ(R1) is
symmetric and singular.
Generalization for R1 = Zp× 2
p
Z ( p is a prime
number):
Lemma 2.1: The number of vertices of G = ( 2
p
Z ) is p – 1
and G = ( 2
p
Z ) is Kp -1, where p is a prime number [4]
Proof: The multiples of p less than p2
are p, 2p, 3p,………,
(p - 1)p. These multiples of p are the only non-zero zero-
divisors of 2
p
Z . If G = ( 2
p
Z ) is the zero-divisor graph
of 2
p
Z , then the vertices of G = ( 2
p
Z ) are the non-zero
zero-divisors of 2
p
Z . So, the vertex set of G = ( 2
p
Z ) is
Z ( 2
p
Z )* and p, 2p, 3p,………,(p - 1)p are the vertices of
G = ( 2
p
Z ). Hence, the number of vertices of G = ( 2
p
Z )
is p – 1. Also, in G = ( 2
p
Z ), every vertex is adjacent to
every other vertex. This gives G = ( 2
p
Z ) is Kp -1.
Theorem 2.2: Let R1 be a finite commutative ring such that
R1 = 2
pp ZZ ( p is a prime number). Let G = (R1) be the
zero-divisor graph with vertex set Z(R1)*. Then number of
vertices of G = (R1) is 2p2
– p – 1, (G) = p2
– 1 and
(G) = p – 1.
Proof: Let R1 be a finite commutative ring such that R1 =
2
pp ZZ (p is a prime number). Let R1*= R1 – {0}. Then
R1* can be partitioned into disjoint sets A, B, C, D and E such
that A = {(u, 0) : u Zp*}, B = {(0, v) : v
*
2
p
Z and v
Z ( 2
p
Z )* }, C = {(0, w) : w
*
2
p
Z and w Z ( 2
p
Z )* },
D = {(a ,b) : a Zp*, b
*
2
p
Z and b Z ( 2
p
Z )* } and E =
{(c ,d) : c Zp*, d
*
2
p
Z and d Z ( 2
p
Z )* }. Clearly, all
the elements in A, B, C are non-zero zero-divisors. Let
(a, b) D and (0, w) C. Here b, w Z ( 2
p
Z )*. So, bp
and wp . This gives bwp2
. Therefore, (a, b) (0, w) =
(0, 0). Hence, every element of D is a non-zero zero-divisor.
But product of any two elements of E is not equal to zero.
Also, product of any element of E with any element of A, B, C
and D is not equal to zero because, cu 0 for c, u Zp*,
dv 0 for d, v
*
2
p
Z and d, v Z ( 2
p
Z )*, dw 0 for d, w
*
2
p
Z and d Z ( 2
p
Z )*, w Z ( 2
p
Z )* and ca 0 for
c, a Zp* respectively. So, no element of E is a non-zero
zero-divisor. Let G = (R1) be the zero-divisor graph with
vertex set Z(R1)*.Then Z(R1)* can be partitioned into four
disjoint sets A, B, C and D. Now using the Lemma 2.1 we
have A= Zp*= p – 1, B =
*
2
p
Z –Z ( 2
p
Z )* =
(p2
–1) – (p – 1) = p2
– 1 – p + 1 = p2
– p, C=
Z ( 2
p
Z )* = p – 1,D= Zp* Z ( 2
p
Z )* = (p – 1)
(p – 1) = p2
– 2p + 1.
Therefore, Z(R1)*=A+B+C+D = (p – 1) +
(p2
– p) + (p – 1) + (p2
– 2p + 1) = 2p2
– p – 1.
So, the number of vertices of G = (R1) is 2p2
– p – 1.
Let s = (u, 0) be any vertex of A.
(i) Every vertex of A is adjacent to every vertex of B. So, s is
adjacent to p2
- p vertices of B.
(ii) Every vertex of A is adjacent to every vertex of C. So, s is
adjacent to p -1 vertices of C.
(iii) Any vertex of A is not adjacent to any vertex of D as ua 0
for u, a Zp*.
Therefore, degG (s) = (p2
- p) + (p -1) = p2
- 1.
Let t = (0, v) be any vertex of B.
(i) Every vertex of B is adjacent to every vertex of A. So, t is
adjacent to p – 1 vertices of A.
(ii) Any vertex of B is not adjacent to any vertex of C as vw 0
for v, w
*
2
p
Z and v Z ( 2
p
Z )*, w Z ( 2
p
Z )*.
(iii) Any vertex of B is not adjacent to any vertex of D as vb 0
for v, b
*
2
p
Z and v Z( 2
p
Z )*, b Z ( 2
p
Z )*.
Therefore, degG (t) = p – 1.
Let x = (0, w) be any vertex of C.
(i) Every vertex of C is adjacent to every vertex of A. So, x is
adjacent to p – 1 vertices of A.
(ii) Any two vertices of C are adjacent to each other. So, x is
adjacent to p – 2 vertices of C.
(iii) Every vertex of C is adjacent to every vertex of D. So, x is
adjacent to p2
- 2p + 1 vertices of D.
(iv) Any vertex of C is not adjacent to any vertex of B as wv 0
for w, v
*
2
p
Z and w Z ( 2
p
Z )*, v Z ( 2
p
Z )*.
Therefore, degG (x) = (p -1) + (p – 2) + (p2
- 2p + 1) = p2
- 2.
Let y = (a, b) be any vertex of D.
(i) Every vertex of D is adjacent to every vertex of C. So, y is
adjacent to p – 1 vertices of C.
(ii) Any vertex of D is not adjacent to any vertex of A as au 0
for u, a Zp*.
(iii) Any vertex of D is not adjacent to any vertex of B as bv 0
for b, v
*
2
p
Z and b Z ( 2
p
Z )*, v Z ( 2
p
Z )*.
Therefore, degG (y) = p -1.
Hence, we have (G) = p2
–1 and (G) = p – 1.
Theorem 2.3: Let M1 be of the adjacency matrix for the zero-
divisor graph G = Γ(R1) of R1 = 2
pp ZZ (p is a prime
number). Then (i) determinant of M1 is zero (ii) M1 is
symmetric and singular.
4. International Journal of Computer Applications Technology and Research
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Proof: Let R1 be a finite commutative ring such that R1 =
2
pp ZZ (p is a prime number). Let G = Γ(R1) be the zero-
divisor graph with vertex set V = Z(R1)* and M1 be the
adjacency matrix for the zero-divisor graph of R1 =
2
pp ZZ .
(i) Since, at least two vertices of G = Γ(R1) are
adjacent to same vertex of G, so M1 contains at least two
identical rows (eg. for Z2×Z4 ). Therefore, the determinant of
the adjacency matrix M1 is zero.
(ii) Clearly M1 is symmetric. Since, the determinant
of the adjacency matrix M1 is zero, M1 is singular.
Theorem 2.4: Let R1 be a finite commutative ring such that R1
= 2
pp ZZ (p is a prime number). Let G = Γ(R1) be the
zero-divisor graph with vertex set V = Z(R1)*.Then nG(V) =
2 (G) – (G), where nG(V) is the neighborhood number,
(G) and (G) denote the maximum and minimum degree
of G respectively.
Proof: Let R1 be a finite commutative ring such that R1 =
2
pp ZZ (p is a prime number). Let G = Γ(R1) be the zero-
divisor graph with vertex set V = Z(R1)*. Since, G = Γ(R1) is
connected [1], we have nG(V) = )(VNG = V =
*
1 )(RZ .
But from Theorem 2.2, we have
*
1 )(RZ = 2p2
– p – 1.
Therefore, nG(V) = 2p2
– p – 1. This implies nG(V) = 2(p2
– 1)
– (p – 1). Also, (G) = p2
– 1 and (G) = p – 1 [from
Theorem 2.2]. This gives nG(V) = 2 (G) – (G).
3. CONSTRUCTION OF ZERO -DIVISOR
GRAPH FOR R2 = pp ZZ 2 ( p IS AN ODD
PRIME NUMBER):
Secondly, we construct the zero-divisor graph for
the ring R2 = pp ZZ 2 (p is an odd prime number) and
analyze the graph. We start with the cases p = 3 and p = 5 and
then generalize the cases.
Case1: When p = 3 we have R2 = Z3×Z6.
The ring R2 has 13 non-zero zero-divisors. In this case V =
Z(R2)* = {(1,0),(2,0),(1,2),(1,3), (1,4), (2,2), (2,3), (2,4), (0,1),
(0,2),(0,3),(0,4),(0,5)} and the zero-divisor graph G = Γ(R2)
is given by:
Fig: 3
The closed neighborhoods of the vertices are
NG[(1,0)] = {(0,1), (0,2), (0,3), (0,4), (0,5), (1,0)}, NG[(2,0)]=
{(0,1), (0,2), (0,3), (0,4),(0,5),(2,0)}, NG[(1,2)] = {(0,3),(1,2)},
NG[(1,3)] = {(0,2), (0,4), (1,3)}, NG[(1,4)] = {(0,3), (1,4)},
NG[(2,2)] = {(0,3), (2,2)}, NG[(2,3)] = {(0,2),(0,4),(2,3)},
NG[(2,4)] = {(0,3), (2,4)}, NG[(0,1)] = {(1,0), (2,0), (0,1)},
NG[(0,2)] = {(1,0), (2,0), (1,3), (2,3), (0,3), (0,2)}, NG[(0,3)] =
{(1,0), (2,0), (1,2),(1,4),(2,2),(2,4),(0,2),(0,4),(0,3)}, NG[(0,4)]
= {(1,0),(2,0),(1,3),(2,3),(0,3),(0,4)}, NG[(0,5)] = {(1,0), (2,0),
(0,5)}.The neighborhood of V is given by NG(V) = {(1,0),
(2,0), (1,2), (1,3), (1,4), (2,2), (2,3), (2,4), (0,1), (0,2), (0,3),
(0,4), (0,5)}.The maximum degree is (G) = 8 and minimum
degree is (G) = 1. The adjacency matrix for the zero-
divisor graph of R2 = Z3×Z6 is M2 ═
13135585
5888
BA
AO
T ,
where A8 5 =
00100
01010
00100
00100
01010
00100
11111
11111
, B5 5 =
00000
00100
01010
00100
00000
,
O88 is the zero matrix and AT
58 is the transpose of A85.
Properties of adjacency matrix M2:
(i) The determinant of the adjacency matrix M2 corresponding
to G = Γ(R2) is 0.
(ii) The rank of the adjacency matrix M2 corresponding to G
= Γ(R2) is 2.
(iii) The adjacency matrix M2 corresponding to G = Γ(R2 ) is
symmetric and singular.
Case2: When p = 5 we have R2 = Z5×Z10.
The ring R2 has 33 non-zero zero-divisors. In this case V =
Z(R2)* = {(1,0),(2,0),(3,0),(4,0), (1,2), (1,4), (1,5), (1,6), (1,8),
(2,2),(2,4),(2,5),(2,6),(2,8),(3,2),(3,4), (3,5), (3,6), (3,8), (4,2),
(4,4), (4,5), (4,6), (4,8), (0,1), (0,2), (0,3), (0,4), (0,5), (0,6),
(0,7), (0,8), (0,9)} and the zero-divisor graph G = Γ(R2) is
given by:
5. International Journal of Computer Applications Technology and Research
Volume 2– Issue 3, 315 - 323, 2013
www.ijcat.com 319
Fig: 4
The closed neighborhoods of the vertices are
NG[(1,0)] = {(0,1), (0,2), (0,3), (0,4), (0,5), (0,6), (0,7), (0,8),
(0,9), (1,0)}, NG[(2,0)] = {(0,1), (0,2), (0,3), (0,4), (0,5), (0,6),
(0,7), (0,8), (0,9), (2,0)}, NG[(3,0)] = {(0,1), (0,2), (0,3), (0,4),
(0,5), (0,6), (0,7), (0,8), (0,9), (3,0)}, NG[(4,0)] = {(0,1), (0,2),
(0,3), (0,4), (0,5), (0,6), (0,7), (0,8), (0,9), (4,0)}, NG[(1,2)] =
{(0,5), (1,2)}, NG[(1,4)] = {(0,5), (1,4)}, NG[(1,5)] = {(0,2),
(0,4), (0,6), (0,8), (1,5)}, NG[(1,6)] = {(0,5), (1,6)}, NG[(1,8)]
= {(0,5),(1,8)}, NG[(2,2)] = {(0,5),(2,2)}, NG[(2,4)] = {(0,5),
(2,4)}}, NG[(2,5)] = {(0,2), (0,4), (0,6), (0,8), (2,5)}, NG[(2,6)]
= {(0,5), (2,6)}, NG[(2,8)] = {(0,5),(2,8)}, NG[(3,2)] = {(0,5),
(3,2)}, NG[(3,4)] = {(0,5), (3,4)}, NG[(3,5)] = {(0,2), (0,4),
(0,6),(0,8),(3,5)}, NG[(3,6)] ={(0,5), (3,6)}, NG[(3,8)] = {(0,5),
(3,8)}, NG[(4,2)] = {(0,5), (4,2)}}, NG[(4,4)] = {(0,5), (4,4)},
NG[(4,5)] = {(0,2), (0,4), (0,6), (0,8), (4,5)}, NG[(4,6)] = {(0,5),
(4,6)}, NG[(4,8)] = {(0,5), (4,8)}, NG[(0,1)] = {(1,0), (2,0),
(3,0), (4,0), (0,1)}, NG[(0,2)] = {(1,0), (2,0), (3,0), (4,0), (0,5),
(1,5), (2,5), (3,5), (4,5), (0,2)}, NG[(0,3)] = {(1,0), (2,0), (3,0),
(4,0),(0,3)}, NG[(0,4)] = {(1,0), (2,0), (3,0), (4,0), (0,5), (1,5),
(2,5), (3,5), (4,5), (0,4)}, NG[(0,5)] = {(1,0), (2,0),(3,0), (4,0),
(1,2), (1,4),(1,6), (1,8), (2,2), (2,4), (2,6), (2,8), (3,2), (3,4),
(3,6), (3,8), (4,2), (4,4), (4,6), (4,8), (0,2), (0,4), (0,6), (0,8),
(0,5)}, NG[(0,6)] = {(1,0),(2,0),(3,0), (4,0), (1,5),(2,5), (3,5),
(4,5), (0,5),(0,6)}, NG[(0,7)] = {(1,0), (2,0), (3,0), (4,0), (0,7)},
NG[(0,8)] = {(1,0), (2,0), (3,0), (4,0), (1,5), (2,5), (3,5), (4,5),
(0,5),(0,8)}, NG[(0,9)] = {(1,0), (2,0), (3,0), (4,0), (0,9)}.The
neighborhood of V is given by NG(V) = {(1,0), (2,0), (3,0),
(4,0), (1,2), (1,4), (1,5), (1,6), (1,8), (2,2), (2,4), (2,5), (2,6),
(2,8), (3,2), (3,4),(3,5),(3,6),(3,8), (4,2), (4,4), (4,5),(4,6), (4,8),
(0,1), (0,2), (0,3), (0,4), (0,5), (0,6), (0,7), (0,8), (0,9)}. The
maximum degree is (G) = 24 and minimum degree is
(G) = 1. The adjacency matrix for the zero-divisor graph of
R2 = Z5×Z10 is M2 ═
333399249
9242424
BA
AO
T ,
A24 9 =
000010000
000010000
010101010
000010000
000010000
000010000
000010000
010101010
000010000
000010000
000010000
000010000
010101010
000010000
000010000
000010000
000010000
010101010
000010000
000010000
111111111
111111111
111111111
111111111
,
B99 =
000000000
000010000
000000000
000010000
010101010
000010000
000000000
000010000
000000000
and
O2424 is the zero matrix and AT
924 is the transpose of A249.
Properties of adjacency matrix M2:
(i) The determinant of the adjacency matrix M2 corresponding
to G = Γ(R2) is 0.
(ii) The rank of the adjacency matrix M2 corresponding to
G = Γ(R2) is 2.
(iii) The adjacency matrix M2 corresponding to G = Γ(R2)
is symmetric and singular.
Generalization for R2 = Zp×Z2p ( p is an odd
prime number):
Lemma 3.1: The number of vertices of G = ( pZ2 ) is p and
G = ( pZ2 ) is K1, p – 1, where p is an odd prime number.
Proof: The multiples of 2 less than 2p are 2, 4, 6, ……,
2(p – 1 ). The non-zero zero-divisors of pZ2 are p and 2, 4,
6, ……,2(p – 1 ). If G = ( pZ2 ) is the zero-divisor graph
of pZ2 , then the vertices of G = ( pZ2 ) are the non-zero
zero-divisors of pZ2 . So, the vertex set of G = ( pZ2 ) is
Z( pZ2 )* and p and 2, 4, 6, ……,2(p – 1 ) are the vertices of
( pZ2 ). Hence, the number of vertices of G = ( pZ2 ) is p.
Also, in G = ( pZ2 ), p is adjacent to remaining vertices 2, 4,
6, ……,2(p – 1 ). This gives G = ( pZ2 ) is K1, p – 1.
Theorem 3.2: Let R2 be a finite commutative ring such that R2 =
pp ZZ 2 (p is an odd prime number). Let G = (R2) be the
zero-divisor graph with vertex set Z(R2)*. Then number of
vertices of G = (R2) is p2
+2p – 2, (G) = p2
–1 and (G) = 1.
6. International Journal of Computer Applications Technology and Research
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www.ijcat.com 320
Proof: Let R2 be a finite commutative ring such that that R2 =
pp ZZ 2 (p is an odd prime number). Let R2* = R2 – {0}.
Then R2* can be partitioned into disjoint sets A, B, C, D and E
such that A = {(u, 0) : u Zp* }, B = {(0, v) : v
*
2 pZ and
v Z ( pZ2 )* }, C = {(0, w) : w
*
2 pZ and w Z ( pZ2 )* },
D = {(a ,b) : a Zp*, b
*
2 pZ and b Z ( pZ2 )* } and E =
{(c ,d) : c Zp*, d
*
2 pZ and d Z ( pZ2 )*} respectively.
Clearly, all the elements in A, B, C are non-zero zero-divisors.
Let (a, b) D. Then (a, b) is of the form either (a, p) or (a, q),
where q = 2m, 1 ≤ m ≤ p – 1. Again let (0, w) C. Similarly,
(0, w) is of the form either (0, p) or (0, q), where q = 2m, 1 ≤ m ≤
p – 1. Now pp and q2 . This gives pqp2 . Therefore,
(a, p) (0, q) = (0, 0) and (a, q) (0, p) = (0, 0). Hence, every
element of D is a non-zero zero-divisor. But product of any two
elements of E is not equal to zero. Also, product of any element
of E with any element of A, B, C and D is not equal to zero
because, cu 0 for c, u Zp*, dv 0 for d, v
*
2 pZ and d,
v Z ( pZ2 )*, dw 0 for d, w
*
2 pZ and d Z ( pZ2 )*,
w Z ( pZ2 )* and ca 0 for c, a Zp* respectively. So, no
element of E is a non-zero zero-divisor. Let G = (R2) be the
zero-divisor graph with vertex set Z(R2)*.Then Z(R2)* can be
partitioned into four disjoint sets A, B, C and D. Now using the
Lemma 3.1 we have A=Zp*= p – 1, B =
*
2 pZ –
Z ( pZ2 )* = (2p-1) – p = p – 1, C= Z ( pZ2 )* = p,
D= Zp* Z ( pZ2 )* = (p – 1) p = p2
– p.
Therefore, Z(R2)*= A+B+C+D= (p – 1) +
(p – 1) + p + (p2
– p) = p2
+2p – 2
So, the number of vertices of G = (R2) is p2
+2p – 2.
Let s = (u, 0) be any vertex of A.
(i) Every vertex of A is adjacent to every vertex of B. So, s is
adjacent to p – 1 vertices of B.
(ii) Every vertex of A is adjacent to every vertex of C. So, s is
adjacent to p vertices of C.
(iii) Any vertex of A is not adjacent to any vertex of D as ua 0
for u, a Zp*.
Therefore, degG (s) = (p - 1) + p = 2p -1.
Let t = (0, v) be any vertex of B.
(i) Every vertex of B is adjacent to every vertex of A. So, t is
adjacent to p – 1 vertices of A
(ii) Any vertex of B is not adjacent to any vertex of C as vw 0
for and v, w
*
2 pZ and v Z ( pZ2 )*, w Z ( pZ2 )*
(iii) Any vertex of B is not adjacent to any vertex of D as vb 0
for v, b
*
2 pZ and v Z ( pZ2 )*, b Z ( pZ2 )*.
Therefore, degG (t) = p - 1
Let x = (0, w) be any vertex of C. Then either x = (0, p) or
x = (0, q), where q = 2m, 1 ≤ m ≤ p – 1
(i) Every vertex of C is adjacent to every vertex of A. So, x is
adjacent to p – 1 vertices of A.
(ii) Case 1: If x = (0, p), then it is adjacent to p - 1 vertices of C.
Case 2: If x = (0, q), then it is adjacent to only one vertex of
C.
(iii) Case 1: If x = (0, p), then it is adjacent to Zp*2
= (p – 1)2
vertices of D.
Case 2: If x = (0, q), then it is adjacent to Zp*= p – 1
vertices of D.
(iv) Any vertex of C is not adjacent to any vertex of B as wv 0
for w, v
*
2 pZ and w Z ( pZ2 )*, v Z ( pZ2 )*.
Therefore, if x = (0, p), then degG (x) = (p - 1) + (p - 1) + (p – 1)2
= p2
– 1 and if x = (0, q), then degG (x) = (p - 1) + 1 + (p - 1) =
2p – 1.
Let y = (a, b) be any vertex of D. Then either y = (a, p) or y =
(a, q), where a Zp* and q = 2m, 1 ≤ m ≤ p – 1
(i) Case 1: If y = (a, p), then it is adjacent to p - 1 vertices of C.
Case 2: If y = (a, q), then it is adjacent to only one vertex of
C.
(ii) Any vertex of D is not adjacent to any vertex of A as au 0
for u, a Zp*
(iii) Any vertex of D is not adjacent to any vertex of B as bv 0
for b, v
*
2 pZ and b Z ( pZ2 )*, v Z ( pZ2 )*.
Therefore, if y = (a, p), then degG (y) = p - 1 and if y = (a, q),
then degG (y) = 1.
Hence, we have (G) = p2
– 1 and (G) = 1.
Theorem 3.3: Let M2 be of the adjacency matrix for the zero-
divisor graph G = Γ(R2) of R2 = pp ZZ 2 (p is an odd prime
number). Then (i) determinant of M2 is zero (ii) M2 is symmetric
and singular.
Proof: Follows from Theorem 2.3.
Theorem 3.4: Let R2 be a finite commutative ring such that
R2 = pp ZZ 2 (p is an odd prime number). Let G = (R2)
be the zero-divisor graph with vertex set V = Z(R2)*.Then
nG(V) = 2p + (G) – (G), where nG(V) is the neighborhood
number, (G) and (G) denote the maximum and minimum
degree of G respectively.
.Proof: Let R2 be a finite commutative ring such that R2 =
pp ZZ 2 (p is an odd prime number). Let G = (R2) be the
zero-divisor graph with vertex set V = Z(R2)*. Since, G = Γ(R2)
is connected [1], we have nG(V) = )(VNG = V =
*
2 )(RZ .
But from Theorem 3.2, we have
*
2 )(RZ = p2
+2p – 2.
Therefore, nG(V) = p2
+2p – 2.This implies nG(V) =2p + (p2
–1)
– 1. Also (G) = p2
– 1 and (G) = 1 [from Theorem 3.2]. This
gives nG(V) = 2p + (G) – (G).
Remark: If p = 2, then R2 = Z2 Z4. So, this case coincides
with the case 1 of section 2.
7. International Journal of Computer Applications Technology and Research
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4. CONSTRUCTION OF ZERO -DIVISOR
GRAPH FOR R3 = 22
pp ZZ ( FOR THAT
ODD PRIME p FOR WHICH p2
– 2 IS A
PRIME NUMBER):
Thirdly, we construct the zero-divisor graph for the
ring R3 =
22
pp ZZ (for that odd prime p for which p2
– 2
is a prime number) and analyze the graph. We start with the
cases p = 3 and p = 5 and then generalize the cases.
Case1: When p = 3 we have R3 = Z3×Z7.
The ring R3 has 8 non-zero zero-divisors. In this case V =
Z(R3)* = {(1,0), (2,0), (0,1), (0,2),(0,3), (0,4), (0,5),(0,6)} and
the zero-divisor graph G = Γ(R3) is given by:
Fig: 5
The closed neighborhoods of the vertices are
NG[(1,0)] = {(0,1), (0,2), (0,3), (0,5), (0,6), (1,0)}, NG[(2,0)] =
{(0,1), (0,2), (0,3), (0,5),(0,6),(2,0)}, NG[(0,1)] = {(1,0), (2,0),
(0,1)}, NG[(0,2)] = {(1,0), (2,0), (0,2)}, NG[(0,3)] = {(1,0),
(2,0), (0,3)}, NG[(0,4)] = {(1,0), (2,0),(0,4)}, NG[(0,5)]={(1,0),
(2,0),(0,5)}, NG[(0,6)] ={(1,0),(2,0),(0,6)}. The neighborhood
of V is given by NG(V) = {(1,0), (2,0), (0,1), (0,2), (0,3), (0,4),
(0,5),(0,6)}.The maximum degree is (G) = 6 and minimum
degree is (G) = 2. The adjacency matrix for the zero-
divisor graph of R3 = Z3×Z7 is M3 ═
886626
6222
OA
AO
T
where
all the entries of 62A is 1, 26
T
A is the transpose of 62A
and 22O , 66O are the zero matrices.
Properties of adjacency matrix M3:
(i) The determinant of the adjacency matrix M3 corresponding
to G = Γ(R3) is 0.
(ii) The rank of the adjacency matrix M3 corresponding to G
= Γ(R3) is 2.
(iii) The adjacency matrix M3 corresponding to G = Γ(R3) is
symmetric and singular.
Case2: When p = 5 we have R3 = Z5×Z23.
The ring R3 has 26 non-zero zero-divisors. In this case V =
Z(R3)* = {(1,0), (2,0), (3,0), (4,0), (0,1),(0,2),(0,3),(0,4), (0,5),
(0,6), (0,7), (0,8), (0,9), (0,10), (0,11), (0,12), (0,13), (0,14),
(0,15), (0,16), (0,17), (0,18), (0,19),(0,20), (0,21), (0,22)}
and the zero-divisor graph G = Γ(R3) is given by:
Fig: 6
The closed neighborhoods of the vertices are
NG[(1,0)] = {(0,1), (0,2), (0,3), (0,4), (0,5), (0,6), (0,7), (0,8),
(0,9), (0,10), (0,11), (0,12), (0,13), (0,14), (0,15), (0,16),
(0,17), (0,18)}, (0,19), (0,20), (0,21),(0,22),(1,0)}, NG[(2,0)] =
{(0,1), (0,2), (0,3), (0,4), (0,5), (0,6), (0,7),(0,8),(0,9), (0,10),
(0,11), (0,12), (0,13), (0,14), (0,15), (0,16), (0,17),(0,18)},
(0,19), (0,20), (0,21), (0,22), (2,0)}, NG[(3,0)] = {(0,1), (0,2),
(0,3), (0,4), (0,5), (0,6), (0,7),(0,8),(0,9), (0,10), (0,11), (0,12),
(0,13), (0,14), (0,15), (0,16), (0,17), (0,18)}, (0,19), (0,20),
(0,21), (0,22), (3,0)}, NG[(4,0)] = {(0,1), (0,2), (0,3), (0,4),
(0,5), (0,6), (0,7), (0,8), (0,9), (0,10), (0,11), (0,12), (0,13),
(0,14), (0,15), (0,16), (0,17), (0,18)}, (0,19), (0,20), (0,21),
(0,22), (4,0)}, NG[(0,1)] = {(1,0), (2,0), (3,0), (4,0), (0,1)},
NG[(0,2)] = {(1,0), (2,0), (3,0), (4,0), (0,2)}, NG[(0,3)] =
{(1,0), (2,0), (3,0), (4,0), (0,3)}, NG[(0,4)] = {(1,0), (2,0),
(3,0),(4,0),(0,4)}, NG[(0,5)] = {(1,0),(2,0),(3,0), (4,0),(0,5)},
NG[(0,6)] = {(1,0), (2,0),(3,0),(4,0),(0,6)}, NG[(0,7)] = {(1,0),
(2,0), (3,0), (4,0), (0,7)}, NG[(0,8)] ={(1,0),(2,0),(3,0),(4,0),
(0,8)}, NG[(0,9)] = {(1,0), (2,0), (3,0),(4,0),(0,9)}, NG[(0,10)]
={(1,0), (2,0), (3,0), (4,0),(0,10)}, NG[(0,11)] = {(1,0), (2,0),
(3,0), (4,0), (0,11)}, NG[(0,12)] = {(1,0),(2,0), (3,0), (4,0),
(0,12)}, NG[(0,13)] = {(1,0), (2,0), (3,0),(4,0),(0,13)},
NG[(0,14)] ={(1,0), (2,0), (3,0), (4,0),(0,14)}, NG[(0,15)] =
{(1,0), (2,0), (3,0), (4,0), (0,15)}, NG[(0,16)] = {(1,0), (2,0),
(3,0), (4,0), (0,16)}, NG[(0,17)] = {(1,0), (2,0), (3,0), (4,0),
(0,17)}, NG[(0,18)] = {(1,0), (2,0), (3,0), (4,0), (0,18)},
NG[(0,19)] = {(1,0), (2,0), (3,0),(4,0),(0,19)}, NG[(0,20)] =
{(1,0), (2,0), (3,0), (4,0), (0,20)}, NG[(0,21)] = {(1,0), (2,0),
(3,0), (4,0), (0,21)}, NG[(0,22)] ={(1,0), (2,0), (3,0), (4,0),
(0,22)}. The neighborhood of V is given by NG(V) = {(1,0),
(2,0), (3,0), (4,0), (0,1), (0,2), (0,3), (0,4), (0,5), (0,6), (0,7),
(0,8), (0,9), (0,10), (0,11), (0,12), (0,13),(0,14), (0,15), (0,16),
(0,17), (0,18), (0,19), (0,20), (0,21),(0,22)}. The maximum
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degree is (G) = 22 and minimum degree is (G) = 4. The
adjacency matrix for the zero-divisor graph of R3 = Z5×Z23 is
M3 ═
26262222422
22444
OA
AO
T where all the entries of
224A is 1, 422
T
A is the transpose of 224A
and 44O , 2222O are the zero matrices.
Properties of adjacency matrix M3:
(i) The determinant of the adjacency matrix M3 corresponding
to G = Γ(R3) is 0.
(ii) The rank of the adjacency matrix M3 corresponding to G
= Γ(R3) is 2.
(iii) The adjacency matrix M3 corresponding to G = Γ(R3) is
symmetric and singular.
Generalization for R3 = 22
pp ZZ (for that
odd prime p for which p2
–2 is a prime
number):
Theorem 4.1: Let R3 be a finite commutative ring such that R3 =
22
pp ZZ (for that odd prime p for which p2
– 2 is a prime
number). Let G = (R3) be the zero-divisor graph with vertex set
Z(R3)*. Then number of vertices of G = (R3) is p2
+ p – 4,
(G) = p2
– 3 and (G) = p – 1.
Proof: Let R3 be a finite commutative ring such that that R3 =
22
pp ZZ (for that odd prime p for which p2
– 2 is a prime
number). Let R3*= R3 – {0}. Then R3* can be partitioned into
disjoint sets A, B and C such that A = {(u, 0): u Zp*}, B =
{(0, v): v 2
* 2
pZ } and C = {(a, b) : a Zp* and
b 2
* 2
pZ } respectively. Clearly, all the elements of A and B
are non-zero zero-divisors. But product of any two elements of
C is not equal to zero. Also, product of any element of C with
any element of A and B is not equal to zero because, au 0 for
a, u Zp*, bv 0 for b, v 2
* 2
pZ respectively. So, no
element of C is a non-zero zero-divisor. Let G = (R3) be the
zero-divisor graph with vertex set Z(R3)*.Then Z(R3)* can be
partitioned into two disjoint sets A and B. Now, A= Zp*=
p – 1 and B= 2
* 2
pZ = p2
– 3. Therefore, Z(R3)* =
A+B= Zp*+ 2
* 2
pZ = (p - 1) + (p2
- 3) = p2
+ p – 4.
So, the number of vertices of G = (R3) is p2
+ p – 4.
Let x = (u, 0) be any vertex of A.
(i) Every vertex of A is adjacent to every vertex of B. So, x is
adjacent to p2
– 3 vertices of B. Therefore, degG (x) = p2
– 3.
Let y = (0, v) be any vertex of B.
(i) Every vertex of B is adjacent to every vertex of A. So, y is
adjacent to p – 1 vertices of A. Therefore, degG (y) = p – 1.
Hence, we have (G) = p2
– 3 and (G) = p – 1.
Theorem 4 .2: Let M3 be of the adjacency matrix for the zero-
divisor graph G = Γ(R3) of R3 =
22
pp ZZ (for that odd
prime p for which p2
– 2 is a prime number). Then (i)
determinant of M3 is zero (ii) M3 is symmetric and singular.
Proof: Follows from Theorem 2.3.
Theorem 4.3: Let R3 be a finite commutative ring such that R3
=
22
pp ZZ (for that odd prime p for which p2
– 2 is a
prime number). Let G = (R3) be the zero-divisor graph with
vertex set V = Z(R3)*. Then nG(V) = (G) + (G), where
nG(V) is the neighborhood number, (G) and (G) denote
the maximum and minimum degree of G respectively.
Proof: Let R3 be a finite commutative ring such that R3
=
22
pp ZZ (for that odd prime p for which p2
– 2 is a prime
number). Let G = (R3) be the zero-divisor graph with vertex set
V = Z(R3)*. Since, G = Γ(R3) is connected [1], we have
nG(V) = )(VNG = V =
*
3 )(RZ . But from Theorem 4.1,
we have
*
3 )(RZ = p2
+p – 4. Therefore, nG(V) = p2
+p – 4.
This implies nG(V) = (p2
– 3) + (p –1). Also, (G) = p2
– 3 and
(G) = p –1 [from Theorem 4.1]. This gives nG(V) = (G) +
(G).
Remark: If p = 2, then R3 = Z2Z2. In this case V = Z(R3)* =
{(0, 1),(1, 0)} and G = (R3) is a 1- regular graph. Also, nG(V)
= 2 = 2 (G) = 2 (G).
5. DEFINITIONS AND RELATIONS:
Let R be a commutative ring with unity and let aR.
Then annihilator of a is denoted by ann(a) and defined by
ann(a) = {x R :ax = 0}. Let ann*(a) = {x ( 0) R: ax = 0}.
The degree of a vertex v of a graph G denoted by
deg(v) is the number of lines incident with v.
Given a zero-divisor graph Γ(R) with vertex set
Z(R)*, then degree of a vertex v of Γ(R) is given by deg(v) =
ann*(v).
Let A and B be two commutative rings with unity.
Then the direct product A B of A and B is also a
commutative ring with unity.
Let G be a graph and V(G) be the vertex set of G.
Let a, b V(G). We define a relation R on V(G) as follows.
For a, b V(G), a is related to b under the relation R if and
only if a and b are not adjacent and for any x V(G), a and x
are adjacent if and only if b and x are adjacent. We denote this
relation by aR b.
6. Results of annihilators on Γ(A B):
Theorem 6.1: The relation R is an equivalence relation on
V(G), where G is any graph.
Proof: For every a V(G), we have aR a, as G has no self-
loop. For a, b V(G), aR b, then clearly, bR a. Again let
9. International Journal of Computer Applications Technology and Research
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www.ijcat.com 323
aRb and bR c. If possible suppose, a and c are adjacent. Then
we have b and c are also adjacent, a contradiction. So, a and c
are not adjacent. Also for x V(G), a and x are adjacent b
and x are adjacent c and x are adjacent. Therefore, aR c.
Hence, the relation R is an equivalence relation on V(G).
Theorem 6.2: For distinct a, bZ (A B)*, aR b in Γ(A B)
if and only if ann(a) – {a} = ann(b) – {b}. Moreover, if aR b
in Γ(R1 R2), then ann(a1) – {a1} = ann(b1) – {b1} and
ann(a2) – {a2} = ann(b2) – {b2}, where a = (a1,a2), b =
(b1,b2), a1,a2 A and b1,b2 B.
Proof: First suppose, for distinct a, b Z(A B)*, aR b in
Γ(A B). Let x ann(a) – {a}. This gives ax = 0, a x. So,
a and x are adjacent. Since aR b we have b and x are adjacent.
Therefore, we have bx = 0, b x. Hence, x ann(b) – {b}.
This implies ann(a) – {a} ann(b) – {b}. Similarly, ann(b) –
{b} ann(a) – {a}. This gives ann(a) – {a} = ann(b) – {b}.
Conversely suppose, ann(a) – {a} = ann(b) – {b}.
Assume that a and b are adjacent. This gives ab = 0
b ann(a) – {a} = ann(b) – {b}, a contradiction. So, a and b
are not adjacent. Again for x Z(A B)*, a and x are adjacent
ax = 0 x ann(b) – {b} bx = 0 b and x are
adjacent. This gives aR b in Γ(A B).
If a = (a1, a2), b = (b1,b2) Z(A B)* , let x ann(a)
– {a}, where x = (x1,x2). Then ax = 0, a x. Therefore, a and
x are adjacent. Since aR b in Γ(A B), we have a and x are
adjacent b and x are adjacent. So, ax = 0 bx = 0. This
gives a1x1 = 0, a2x2 = 0 b1x1 = 0, b2x2 = 0. Hence we have
a1x1 = 0 b1x1 = 0, (x1 a1,b1) in A and a2x2 = 0 b2x2 = 0
(x2 a2,b2) in B. Therefore, ann(a1) – {a1} = ann(b1) – {b1}
and ann(a2) – {a2} = ann(b2) – {b2}.
Example 6.3: Consider the commutative ring Z2×Z4 = {(0,0),
(0,1), (0,2), (0,3), (1,0), (1,1), (1,2), (1,3)} and zero-divisor
graph Γ(Z2×Z4). Here Z(Z2×Z4)* = {(0,1), (0,2), (0,3), (1,0),
(1,2)}.The possible edges are {(0, 1), (1,0)}, {(0,2), (1,0)},
{(0,3), (1,0)} and {(0,2), (1,2)}. The pairs {(0, 1), (0, 2)},
{(0, 1), (0, 3)}, {(0, 2), (0, 3)} and {(1, 0), (1, 2)} establish the
existence of relation R and Theorem 6.1:
7. CONCLUSIONS:
In this paper, we study the adjacency matrix and
neighborhood associated with zero-divisor graph for direct
product of finite commutative rings. Neighborhoods may be
used to represent graphs in computer algorithms, via the
adjacency list and adjacency matrix representations.
Neighborhoods are also used in the clustering coefficient of a
graph, which is a measure of the average density of its
neighborhoods. In addition, many important classes of graphs
may be defined by properties of their neighborhoods.
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