In theoretical chemistry, the researchers use graph models to express the structure of molecular, and the Zagreb indices and multiplicative Zagreb indices defined on molecular graph G are applied to measure the chemical characteristics of compounds and drugs. In this paper, we present the exact expressions of multiplicative Zagreb indices for certain important chemical structures like nanotube, nanostar and polyomino chain.
{"title":"The Multiplicative Zagreb Indices of Nanostructures and Chains","authors":"Wei Gao, M. Farahani, M. Kanna","doi":"10.4236/OJDM.2016.62008","DOIUrl":"https://doi.org/10.4236/OJDM.2016.62008","url":null,"abstract":"In theoretical chemistry, the researchers use graph models to express the structure of molecular, and the Zagreb indices and multiplicative Zagreb indices defined on molecular graph G are applied to measure the chemical characteristics of compounds and drugs. In this paper, we present the exact expressions of multiplicative Zagreb indices for certain important chemical structures like nanotube, nanostar and polyomino chain.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"06 1","pages":"82-88"},"PeriodicalIF":0.0,"publicationDate":"2016-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626856","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}
We study the mixing rate of non-backtracking random walks on graphs by looking at non-backtracking walks as walks on the directed edges of a graph. A result known as Ihara’s Theorem relates the adjacency matrix of a graph to a matrix related to non-backtracking walks on the directed edges. We prove a weighted version of Ihara’s Theorem which relates the transition probability matrix of a non-backtracking walk to the transition matrix for the usual random walk. This allows us to determine the spectrum of the transition probability matrix of a non-backtracking random walk in the case of regular graphs and biregular graphs. As a corollary, we obtain a result of Alon et al. in [1] that in most cases, a non-backtracking random walk on a regular graph has a faster mixing rate than the usual random walk. In addition, we obtain an analogous result for biregular graphs.
{"title":"Non-backtracking random walks and a weighted Ihara's theorem","authors":"Mark Kempton","doi":"10.4236/OJDM.2016.64018","DOIUrl":"https://doi.org/10.4236/OJDM.2016.64018","url":null,"abstract":"We study the mixing rate of non-backtracking random walks on graphs by looking at non-backtracking walks as walks on the directed edges of a graph. A result known as Ihara’s Theorem relates the adjacency matrix of a graph to a matrix related to non-backtracking walks on the directed edges. We prove a weighted version of Ihara’s Theorem which relates the transition probability matrix of a non-backtracking walk to the transition matrix for the usual random walk. This allows us to determine the spectrum of the transition probability matrix of a non-backtracking random walk in the case of regular graphs and biregular graphs. As a corollary, we obtain a result of Alon et al. in [1] that in most cases, a non-backtracking random walk on a regular graph has a faster mixing rate than the usual random walk. In addition, we obtain an analogous result for biregular graphs.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"531 1","pages":"207-226"},"PeriodicalIF":0.0,"publicationDate":"2016-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626510","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}
The bipartite Star123-free graphs were introduced by V. Lozin in [1] to generalize some already known classes of bipartite graphs. In this paper, we extend to bipartite Star123-free graphs a linear time algorithm of J. L. Fouquet, V. Giakoumakis and J. M. Vanherpe for finding a maximum matching in bipartite Star123, P7-free graphs presented in [2]. Our algorithm is a solution of Lozin’s conjecture.
V. Lozin在2010年引入了二部图Star123-free，推广了一些已知的二部图类。本文将J. L. Fouquet, V. Giakoumakis和J. M. Vanherpe在[2]中提出的寻找二部Star123, P7-free图的最大匹配的线性时间算法推广到二部Star123-free图。我们的算法是Lozin猜想的一个解。
{"title":"Solving the Maximum Matching Problem on Bipartite Star123-Free Graphs in Linear Time","authors":"Ruzayn Quaddoura","doi":"10.4236/OJDM.2016.61003","DOIUrl":"https://doi.org/10.4236/OJDM.2016.61003","url":null,"abstract":"The \u0000bipartite Star123-free \u0000graphs were introduced by V. Lozin in [1] to generalize some already known classes of \u0000bipartite graphs. In this paper, we extend to bipartite Star123-free graphs a linear time algorithm of J. L. \u0000Fouquet, V. Giakoumakis and J. M. Vanherpe for finding a maximum matching in \u0000bipartite Star123, P7-free graphs presented in [2]. Our algorithm is a solution of Lozin’s \u0000conjecture.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"41 1","pages":"13-24"},"PeriodicalIF":0.0,"publicationDate":"2016-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626541","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}
A decomposition of a graph H is a partition of the edge set of H into edge-disjoint subgraphs . If for all , then G is a decomposition of H by G. Two decompositions and of the complete bipartite graph are orthogonal if, for all . A set of decompositions of is a set of k mutually orthogonal graph squares (MOGS) if and are orthogonal for all and . For any bipartite graph G with n edges, denotes the maximum number k in a largest possible set of MOGS of by G. Our objective in this paper is to compute where is a path of length d with d + 1 vertices (i.e. Every edge of this path is one-to-one corresponding to an isomorphic to a certain graph F).
{"title":"On Mutually Orthogonal Graph-Path Squares","authors":"R. El-Shanawany","doi":"10.4236/OJDM.2016.61002","DOIUrl":"https://doi.org/10.4236/OJDM.2016.61002","url":null,"abstract":"A decomposition of a graph H is a partition of the edge set of H into edge-disjoint subgraphs . If for all , then G is a decomposition of H by G. Two decompositions and of the complete bipartite graph are orthogonal if, for all . A set of decompositions of is a set of k mutually orthogonal graph squares (MOGS) if and are orthogonal for all and . For any bipartite graph G with n edges, denotes the maximum number k in a largest possible set of MOGS of by G. Our objective in this paper is to compute where is a path of length d with d + 1 vertices (i.e. Every edge of this path is one-to-one corresponding to an isomorphic to a certain graph F).","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"51 1","pages":"7-12"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626480","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}
The question associated with total domination on the queen’s graph has a long and rich history, first having been posed by Ahrens in 1910 [1]. The question is this: What is the minimum number of queens needed so that every square of an n × n board is attacked? Beginning in 2005 with Amirabadi, Burchett, and Hedetniemi [2] [3], work on this problem, and two other related problems, has seen progress. Bounds have been given for the values of all three domination parameters on the queen’s graph. In this paper, formations of queens are given that provide new bounds for the values of total, paired, and connected domination on the queen’s graph, denoted , , and respectively. For any n × n board size, the new bound of is arrived at, along with the separate bounds of , for with , and , for with .
与女王图谱上的完全统治有关的问题有着悠久而丰富的历史，最早是由阿伦斯在1910年提出的。问题是这样的:n × n棋盘上的每个方格都被攻击所需的最小皇后数是多少?从2005年开始，Amirabadi、Burchett和Hedetniemi对这个问题以及其他两个相关问题的研究取得了进展。给出了皇后图上所有三个控制参数的取值范围。本文给出了女王的构形，为女王图上的总统治、配对统治和连接统治的值提供了新的界，分别记为、和。对于任何n × n大小的板，到达的新边界，以及单独的边界，For with, and, For with。
{"title":"Paired, Total, and Connected Domination on the Queen’s Graph Revisited","authors":"Paul A. Burchett","doi":"10.4236/OJDM.2016.61001","DOIUrl":"https://doi.org/10.4236/OJDM.2016.61001","url":null,"abstract":"The question associated with total domination on the queen’s graph has a long and rich history, first having been posed by Ahrens in 1910 [1]. The question is this: What is the minimum number of queens needed so that every square of an n × n board is attacked? Beginning in 2005 with Amirabadi, Burchett, and Hedetniemi [2] [3], work on this problem, and two other related problems, has seen progress. Bounds have been given for the values of all three domination parameters on the queen’s graph. In this paper, formations of queens are given that provide new bounds for the values of total, paired, and connected domination on the queen’s graph, denoted , , and respectively. For any n × n board size, the new bound of is arrived at, along with the separate bounds of , for with , and , for with .","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"06 1","pages":"1-6"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626471","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}
Fuzzy greedoids were recently introduced as a fuzzy set generalization of (crisp) greedoids. We characterize fuzzy languages which define fuzzy greedoids, give necessary properties and sufficient properties of the fuzzy rank function of a fuzzy greedoid, give a characterization of the rank function for a weighted greedoid, and discuss the rank closure of a fuzzy greedoid.
{"title":"Rank Functions of Fuzzy Greedoids","authors":"S. Tedford","doi":"10.4236/OJDM.2015.54006","DOIUrl":"https://doi.org/10.4236/OJDM.2015.54006","url":null,"abstract":"Fuzzy greedoids were recently introduced as a fuzzy set generalization of (crisp) greedoids. We characterize fuzzy languages which define fuzzy greedoids, give necessary properties and sufficient properties of the fuzzy rank function of a fuzzy greedoid, give a characterization of the rank function for a weighted greedoid, and discuss the rank closure of a fuzzy greedoid.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"05 1","pages":"65-73"},"PeriodicalIF":0.0,"publicationDate":"2015-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626195","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}
Let G = (V, E) be a simple graph. A set S E(G) is an edge-vertex dominating set of G (or simply an ev-dominating set), if for all vertices v V(G); there exists an edge eS such that e dominates v. Let denote the family of all ev-dominating sets of with cardinality i. Let . In this paper, we obtain a recursive formula for . Using this recursive formula, we construct the polynomial, , which we call edge-vertex domination polynomial of (or simply an ev-domination polynomial of ) and obtain some properties of this polynomial.
设G = (V, E)为简单图。如果对于所有顶点v v (G)，则集合S E(G)是G的边-顶点控制集(或简单地说是ev控制集);存在一条边e，使得e优于v。令表示所有具有基数i的e -支配集合的族。在本文中，我们得到了。利用这个递归公式，构造了一个多项式，我们称之为的边-顶点控制多项式(或简称为ev-控制多项式)，并得到了该多项式的一些性质。
{"title":"Edge-Vertex Dominating Sets and Edge-Vertex Domination Polynomials of Cycles","authors":"A. Vijayan, J. Beula","doi":"10.4236/OJDM.2015.54007","DOIUrl":"https://doi.org/10.4236/OJDM.2015.54007","url":null,"abstract":"Let G = (V, E) be a simple graph. A set S E(G) is an edge-vertex \u0000dominating set of G (or simply an ev-dominating set), if for all vertices v V(G); there exists an \u0000edge eS such that e dominates v. Let denote the family of all ev-dominating sets of with cardinality i. Let . In this paper, we \u0000obtain a recursive formula for . Using this \u0000recursive formula, we construct the polynomial, , which we call edge-vertex domination polynomial of (or simply an ev-domination polynomial of ) and obtain some \u0000properties of this polynomial.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"05 1","pages":"74-87"},"PeriodicalIF":0.0,"publicationDate":"2015-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626397","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}
A set is a dominating set of G if every vertex of is adjacent to at least one vertex of S. The cardinality of the smallest dominating set of G is called the domination number of G. The square G2 of a graph G is obtained from G by adding new edges between every two vertices having distance 2 in G. In this paper we study the domination number of square of graphs, find a bound for domination number of square of Cartesian product of cycles, and find the exact value for some of them.
{"title":"Domination Number of Square of Cartesian Products of Cycles","authors":"M. Alishahi, Sakineh Hoseini Shalmaee","doi":"10.4236/OJDM.2015.54008","DOIUrl":"https://doi.org/10.4236/OJDM.2015.54008","url":null,"abstract":"A set is a dominating set of G if every vertex of is adjacent to at least one vertex of S. The cardinality of the smallest \u0000dominating set of G is called the \u0000domination number of G. The square G2 of a graph G is obtained from G by adding new edges between every two vertices having distance 2 \u0000in G. In this paper we study the \u0000domination number of square of graphs, find a bound for domination number of \u0000square of Cartesian product of cycles, and find the exact value for some of \u0000them.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"05 1","pages":"88-94"},"PeriodicalIF":0.0,"publicationDate":"2015-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626410","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}
A sentence over a finite alphabet A, is a finite sequence of non-empty words over A. More generally, we define a graphical sentence over A by attaching a non-empty word over A to each arrow and each loop of a connected directed graph (digraph, for short). Each word is written according to the direction of its corresponding arrow or loop. Graphical sentences can be used to encode sets of sentences in a compact way: the readable sentences of a graphical sentence being the sentences corresponding to directed paths in the digraph. We apply combinatorial equations on enriched trees and rooted trees, in the context of combinatorial species and Polya theories, to analyze parameters in classes of tree-like sentences. These are graphical sentences constructed on tree-like digraphs.
{"title":"A Combinatorial Analysis of Tree-Like Sentences","authors":"G. Labelle, Louise Laforest","doi":"10.4236/OJDM.2015.53004","DOIUrl":"https://doi.org/10.4236/OJDM.2015.53004","url":null,"abstract":"A sentence over a finite alphabet A, is a finite sequence of non-empty words over A. More generally, we define a graphical sentence over A by attaching a non-empty word over A to each arrow and each loop of a connected directed graph (digraph, for short). Each word is written according to the direction of its corresponding arrow or loop. Graphical sentences can be used to encode sets of sentences in a compact way: the readable sentences of a graphical sentence being the sentences corresponding to directed paths in the digraph. We apply combinatorial equations on enriched trees and rooted trees, in the context of combinatorial species and Polya theories, to analyze parameters in classes of tree-like sentences. These are graphical sentences constructed on tree-like digraphs.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"05 1","pages":"32-53"},"PeriodicalIF":0.0,"publicationDate":"2015-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626120","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}
Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A function is called a signed dominating function (SDF) if for each vertex . The weight of f is defined by . The signed domination number of a digraph D is . Let Cm × Cn denotes the cartesian product of directed cycles of length m and n. In this paper, we determine the exact values of gs(Cm × Cn) for m = 8, 9, 10 and arbitrary n. Also, we give the exact value of gs(Cm × Cn) when m, (mod 3) and bounds for otherwise.
{"title":"On the Signed Domination Number of the Cartesian Product of Two Directed Cycles","authors":"Ramy S. Shaheen","doi":"10.4236/OJDM.2015.53005","DOIUrl":"https://doi.org/10.4236/OJDM.2015.53005","url":null,"abstract":"Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A function is called a signed dominating function (SDF) if for each vertex . The weight of f is defined by . The signed domination number of a digraph D is . Let Cm × Cn denotes the cartesian product of directed cycles of length m and n. In this paper, we determine the exact values of gs(Cm × Cn) for m = 8, 9, 10 and arbitrary n. Also, we give the exact value of gs(Cm × Cn) when m, (mod 3) and bounds for otherwise.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"24 1","pages":"54-64"},"PeriodicalIF":0.0,"publicationDate":"2015-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626183","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}
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