Pub Date : 2019-05-07DOI: 10.13069/JACODESMATH.561322
A. R. M. Hamzekolaee, Y. Talebi
Let $R$ be a ring and $M$ a right $R$-module. Let $N$ be a proper submodule of $M$. We say that $M$ is $N$-coretractable (or $M$ is coretractable relative to $N$) provided that, for every proper submodule $K$ of $M$ containing $N$, there is a nonzero homomorphism $f:M/Krightarrow M$. We present some conditions that a module $M$ is coretractable if and only if $M$ is coretractable relative to a submodule $N$. We also provide some examples to illustrate special cases.
{"title":"Coretractable modules relative to a submodule","authors":"A. R. M. Hamzekolaee, Y. Talebi","doi":"10.13069/JACODESMATH.561322","DOIUrl":"https://doi.org/10.13069/JACODESMATH.561322","url":null,"abstract":"Let $R$ be a ring and $M$ a right $R$-module. Let $N$ be a proper submodule of $M$. We say that $M$ is $N$-coretractable (or $M$ is coretractable relative to $N$) provided that, for every proper submodule $K$ of $M$ containing $N$, there is a nonzero homomorphism $f:M/Krightarrow M$. We present some conditions that a module $M$ is coretractable if and only if $M$ is coretractable relative to a submodule $N$. We also provide some examples to illustrate special cases.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46446303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-05-07DOI: 10.13069/JACODESMATH.560410
Jung-Chao Ban, Chih-Hung Chang
This paper investigates the coloring problem on Fibonacci-Cayley tree, which is a Cayley graph whose vertex set is the Fibonacci sequence. More precisely, we elucidate the complexity of shifts of finite type defined on Fibonacci-Cayley tree via an invariant called entropy. We demonstrate that computing the entropy of a Fibonacci tree-shift of finite type is equivalent to studying a nonlinear recursive system and reveal an algorithm for the computation. What is more, the entropy of a Fibonacci tree-shift of finite type is the logarithm of the spectral radius of its corresponding matrix. We apply the result to neural networks defined on Fibonacci-Cayley tree, which reflect those neural systems with neuronal dysfunction. Aside from demonstrating a surprising phenomenon that there are only two possibilities of entropy for neural networks on Fibonacci-Cayley tree, we address the formula of the boundary in the parameter space.
{"title":"Complexity of neural networks on Fibonacci-Cayley tree","authors":"Jung-Chao Ban, Chih-Hung Chang","doi":"10.13069/JACODESMATH.560410","DOIUrl":"https://doi.org/10.13069/JACODESMATH.560410","url":null,"abstract":"This paper investigates the coloring problem on Fibonacci-Cayley tree, which is a Cayley graph whose vertex set is the Fibonacci sequence. More precisely, we elucidate the complexity of shifts of finite type defined on Fibonacci-Cayley tree via an invariant called entropy. We demonstrate that computing the entropy of a Fibonacci tree-shift of finite type is equivalent to studying a nonlinear recursive system and reveal an algorithm for the computation. What is more, the entropy of a Fibonacci tree-shift of finite type is the logarithm of the spectral radius of its corresponding matrix. We apply the result to neural networks defined on Fibonacci-Cayley tree, which reflect those neural systems with neuronal dysfunction. Aside from demonstrating a surprising phenomenon that there are only two possibilities of entropy for neural networks on Fibonacci-Cayley tree, we address the formula of the boundary in the parameter space.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42014471","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-19DOI: 10.13069/JACODESMATH.508983
Sunil Kumar Raghavan Unnithan, K. Balakrishnan
Betweenness centrality measures the potential or power of a node to control the communication over the network under the assumption that information flows primarily over the shortest paths between pair of nodes. The removal of a node with highest betweenness from the network will most disrupt communications between other nodes because it lies on the largest number of paths. A large network can be thought of as inter-connection between smaller networks by means of different graph operations. Thus the structure of a composite graph can be studied by analysing its component graphs. In this paper we present the betweenness centrality of some classes of composite graphs constructed by the graph operation called amalgamation or merging.
{"title":"Betweenness centrality in convex amalgamation of graphs","authors":"Sunil Kumar Raghavan Unnithan, K. Balakrishnan","doi":"10.13069/JACODESMATH.508983","DOIUrl":"https://doi.org/10.13069/JACODESMATH.508983","url":null,"abstract":"Betweenness centrality measures the potential or power of a node to control the communication over the network under the assumption that information flows primarily over the shortest paths between pair of nodes. The removal of a node with highest betweenness from the network will most disrupt communications between other nodes because it lies on the largest number of paths. A large network can be thought of as inter-connection between smaller networks by means of different graph operations. Thus the structure of a composite graph can be studied by analysing its component graphs. In this paper we present the betweenness centrality of some classes of composite graphs constructed by the graph operation called amalgamation or merging.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44618581","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-19DOI: 10.13069/JACODESMATH.508968
N. Aydin, Derek Foret
Explicit construction of linear codes with best possible parameters is one of the major and challenging problems in coding theory. Cyclic codes and their various generalizations, such as quasi-twisted (QT) codes, are known to contain many codes with best known parameters. Despite the fact that these classes of codes have been extensively searched, we have been able to refine existing search algorithms to discover many new linear codes over the alphabets $mathbb{F}_{3}$, $mathbb{F}_{11}$, and $mathbb{F}_{13}$ with better parameters. A total of 38 new linear codes over the three alphabets are presented.
{"title":"New Linear Codes over $GF(3)$, $GF(11)$, and $GF(13)$","authors":"N. Aydin, Derek Foret","doi":"10.13069/JACODESMATH.508968","DOIUrl":"https://doi.org/10.13069/JACODESMATH.508968","url":null,"abstract":"Explicit construction of linear codes with best possible parameters is one of the major and challenging problems in coding theory. Cyclic codes and their various generalizations, such as quasi-twisted (QT) codes, are known to contain many codes with best known parameters. Despite the fact that these classes of codes have been extensively searched, we have been able to refine existing search algorithms to discover many new linear codes over the alphabets $mathbb{F}_{3}$, $mathbb{F}_{11}$, and $mathbb{F}_{13}$ with better parameters. A total of 38 new linear codes over the three alphabets are presented.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48881060","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-19DOI: 10.13069/JACODESMATH.514339
Ismail Aydogdu
{In this paper we generalize $mathbb{Z}_{2}mathbb{Z}_{2}[u]$-linear codes to codes over $mathbb{Z}_{p}[u]/{langle u^r rangle}timesmathbb{Z}_{p}[u]/{langle u^s rangle}$ where $p$ is a prime number and $u^r=0=u^s$. We will call these family of codes as $mathbb{Z}_{p}[u^r,u^s]$-linear codes which are actually special submodules. We determine the standard forms of the generator and parity-check matrices of these codes. Furthermore, for the special case $p=2$, we define a Gray map to explore the binary images of $mathbb{Z}_{2}[u^r,u^s]$-linear codes. Finally, we study the structure of self-dual $mathbb{Z}_{2}[u^2,u^3]$-linear codes and present some examples.
{"title":"Codes over $mathbb{Z}_{p}[u]/{langle u^r rangle}timesmathbb{Z}_{p}[u]/{langle u^s rangle}$","authors":"Ismail Aydogdu","doi":"10.13069/JACODESMATH.514339","DOIUrl":"https://doi.org/10.13069/JACODESMATH.514339","url":null,"abstract":"{In this paper we generalize $mathbb{Z}_{2}mathbb{Z}_{2}[u]$-linear codes to codes over $mathbb{Z}_{p}[u]/{langle u^r rangle}timesmathbb{Z}_{p}[u]/{langle u^s rangle}$ where $p$ is a prime number and $u^r=0=u^s$. We will call these family of codes as $mathbb{Z}_{p}[u^r,u^s]$-linear codes which are actually special submodules. We determine the standard forms of the generator and parity-check matrices of these codes. Furthermore, for the special case $p=2$, we define a Gray map to explore the binary images of $mathbb{Z}_{2}[u^r,u^s]$-linear codes. Finally, we study the structure of self-dual $mathbb{Z}_{2}[u^2,u^3]$-linear codes and present some examples.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45181245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-19DOI: 10.13069/JACODESMATH.505364
Manjit Singh, Sudhir Batra
Let $mathbb{F}_q$ be a finite field with $q$ elements and $n$ be a positive integer. In this paper, we determine the weight distribution of a class cyclic codes of length $2^n$ over $mathbb{F}_q$ whose parity check polynomials are either binomials or trinomials with $2^l$ zeros over $mathbb{F}_q$, where integer $lge 1$. In addition, constant weight and two-weight linear codes are constructed when $qequiv3pmod 4$.
{"title":"Weight distribution of a class of cyclic codes of length $2^n$","authors":"Manjit Singh, Sudhir Batra","doi":"10.13069/JACODESMATH.505364","DOIUrl":"https://doi.org/10.13069/JACODESMATH.505364","url":null,"abstract":"Let $mathbb{F}_q$ be a finite field with $q$ elements and $n$ be a positive integer. In this paper, we determine the weight distribution of a class cyclic codes of length $2^n$ over $mathbb{F}_q$ whose parity check polynomials are either binomials or trinomials with $2^l$ zeros over $mathbb{F}_q$, where integer $lge 1$. In addition, constant weight and two-weight linear codes are constructed when $qequiv3pmod 4$.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43528459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-12-05DOI: 10.13069/jacodesmath.1056511
Jon-Lark Kim, D. Ohk, Doo Young Park, Jae Woo Park
Choi Seok-Jeong studied Latin squares at least 60 years earlier than Euler although this was less known. He introduced a pair of orthogonal Latin squares of order 9 in his book. Interestingly, his two orthogonal non-double-diagonal Latin squares produce a magic square of order 9, whose theoretical reason was not studied. There have been a few studies on Choi’s Latin squares of order 9. The most recent one is Ko-Wei Lih’s construction of Choi’s Latin squares of order 9 based on the two 3ˆ3 orthogonal Latin squares. In this paper, we give a new generalization of Choi’s orthogonal Latin squares of order 9 to orthogonal Latin squares of size n2 using the Kronecker product including Lih’s construction. We find a geometric description of Choi’s orthogonal Latin squares of order 9 using the dihedral group D8. We also give a new way to construct magic squares from two orthogonal non-double-diagonal Latin squares, which explains why Choi’s Latin squares produce a magic square of order 9.
{"title":"Recent results on Choi's orthogonal Latin squares","authors":"Jon-Lark Kim, D. Ohk, Doo Young Park, Jae Woo Park","doi":"10.13069/jacodesmath.1056511","DOIUrl":"https://doi.org/10.13069/jacodesmath.1056511","url":null,"abstract":"Choi Seok-Jeong studied Latin squares at least 60 years earlier than Euler although this was less known. He introduced a pair of orthogonal Latin squares of order 9 in his book. Interestingly, his two orthogonal non-double-diagonal Latin squares produce a magic square of order 9, whose theoretical reason was not studied. There have been a few studies on Choi’s Latin squares of order 9. The most recent one is Ko-Wei Lih’s construction of Choi’s Latin squares of order 9 based on the two 3ˆ3 orthogonal Latin squares. In this paper, we give a new generalization of Choi’s orthogonal Latin squares of order 9 to orthogonal Latin squares of size n2 using the Kronecker product including Lih’s construction. We find a geometric description of Choi’s orthogonal Latin squares of order 9 using the dihedral group D8. We also give a new way to construct magic squares from two orthogonal non-double-diagonal Latin squares, which explains why Choi’s Latin squares produce a magic square of order 9.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66233310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-09-08DOI: 10.13069/jacodesmath.451229
K. N. Rajeswari, Umesh Gupta
Let $R$ be any ring with identity. An element $a in R$ is called nil-clean, if $a=e+n$ where $e$ is an idempotent element and $n$ is a nil-potent element. In this paper we give necessary and sufficient conditions for a $2times 2$ matrix over an integral domain $R$ to be nil-clean.
{"title":"Characterization of $2times 2$ nil-clean matrices over integral domains","authors":"K. N. Rajeswari, Umesh Gupta","doi":"10.13069/jacodesmath.451229","DOIUrl":"https://doi.org/10.13069/jacodesmath.451229","url":null,"abstract":"Let $R$ be any ring with identity. An element $a in R$ is called nil-clean, if $a=e+n$ where $e$ is an idempotent element and $n$ is a nil-potent element. In this paper we give necessary and sufficient conditions for a $2times 2$ matrix over an integral domain $R$ to be nil-clean.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42405328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-09-08DOI: 10.13069/JACODESMATH.458240
Syed Ahtsham ul Haq Bokhary, Tanveer Iqbal, Usman Ali
The game chromatic number $chi_g$ is investigated for Cartesian product $Gsquare H$ and corona product $Gcirc H$ of two graphs $G$ and $H$. The exact values for the game chromatic number of Cartesian product graph of $S_{3}square S_{n}$ is found, where $S_n$ is a star graph of order $n+1$. This extends previous results of Bartnicki et al. [1] and Sia [5] on the game chromatic number of Cartesian product graphs. Let $P_m$ be the path graph on $m$ vertices and $C_n$ be the cycle graph on $n$ vertices. We have determined the exact values for the game chromatic number of corona product graphs $P_{m}circ K_{1}$ and $P_{m}circ C_{n}$.
研究了两个图$G$和$H$的笛卡尔积$Gsquare H$和电晕积$Gcirc H$的游戏色数$chi_g$。求出了$S_{3}square S_{n}$笛卡尔积图的游戏色数的精确值,其中$S_n$是阶$n+1$的星图。这扩展了Bartnicki et al.[1]和Sia[5]关于笛卡尔积图的博弈色数的先前结果。设$P_m$为$m$顶点上的路径图,$C_n$为$n$顶点上的循环图。我们已经确定了电晕积图$P_{m}circ K_{1}$和$P_{m}circ C_{n}$的游戏色数的确切值。
{"title":"Game chromatic number of Cartesian and corona product graphs","authors":"Syed Ahtsham ul Haq Bokhary, Tanveer Iqbal, Usman Ali","doi":"10.13069/JACODESMATH.458240","DOIUrl":"https://doi.org/10.13069/JACODESMATH.458240","url":null,"abstract":"The game chromatic number $chi_g$ is investigated for Cartesian product $Gsquare H$ and corona product $Gcirc H$ of two graphs $G$ and $H$. The exact values for the game chromatic number of Cartesian product graph of $S_{3}square S_{n}$ is found, where $S_n$ is a star graph of order $n+1$. This extends previous results of Bartnicki et al. [1] and Sia [5] on the game chromatic number of Cartesian product graphs. Let $P_m$ be the path graph on $m$ vertices and $C_n$ be the cycle graph on $n$ vertices. We have determined the exact values for the game chromatic number of corona product graphs $P_{m}circ K_{1}$ and $P_{m}circ C_{n}$.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41980859","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-09-08DOI: 10.13069/JACODESMATH.451218
H. Prodinger
{"title":"Finite Rogers-Ramanujan type continued fractions","authors":"H. Prodinger","doi":"10.13069/JACODESMATH.451218","DOIUrl":"https://doi.org/10.13069/JACODESMATH.451218","url":null,"abstract":"","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47218816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}