The edge-degree distance of a simple connected graph G is defined as the sum of the terms (d(e|G)+d(f|G))d(e,f|G) over all unordered pairs {e,f} of edges of G, where d(e|G) and d(e,f|G) denote the degree of the edge e in G and the distance between the edges e and f in G, respectively. In this paper, we study the behavior of two versions of the edge-degree distance under two graph products called splice and link.
{"title":"Splices, Links, and their Edge-Degree Distances","authors":"M. Azari, Hojjatollah Divanpour","doi":"10.22108/TOC.2017.21614","DOIUrl":"https://doi.org/10.22108/TOC.2017.21614","url":null,"abstract":"The edge-degree distance of a simple connected graph G is defined as the sum of the terms (d(e|G)+d(f|G))d(e,f|G) over all unordered pairs {e,f} of edges of G, where d(e|G) and d(e,f|G) denote the degree of the edge e in G and the distance between the edges e and f in G, respectively. In this paper, we study the behavior of two versions of the edge-degree distance under two graph products called splice and link.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"6 1","pages":"29-42"},"PeriodicalIF":0.4,"publicationDate":"2017-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44548120","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}
Xiaomin Zhu, Lihua Feng, Minmin Liu, Weijun Liu, Yuqin Hu
In this paper, by using the degree sequences of graphs, we present sufficient conditions for a graph to be Hamiltonian, traceable, Hamilton-connected or $k$-connected in light of numerous topological indices such as the eccentric connectivity index, the eccentric distance sum, the connective eccentricity index.
{"title":"Some topological indices and graph properties","authors":"Xiaomin Zhu, Lihua Feng, Minmin Liu, Weijun Liu, Yuqin Hu","doi":"10.22108/TOC.2017.21467","DOIUrl":"https://doi.org/10.22108/TOC.2017.21467","url":null,"abstract":"In this paper, by using the degree sequences of graphs, we present sufficient conditions for a graph to be Hamiltonian, traceable, Hamilton-connected or $k$-connected in light of numerous topological indices such as the eccentric connectivity index, the eccentric distance sum, the connective eccentricity index.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"6 1","pages":"51-65"},"PeriodicalIF":0.4,"publicationDate":"2017-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46166700","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}
In this paper, we extend upon the results of B. Suceavă and R. Stong [Amer. Math. Monthly, 110 (2003) 162–162], which they computed the minimum number of 3-cycles needed to generate an even permutation. Let Ωk,m be the set of all permutations of the form c1c2 · · · ck where ci’s are arbitrary m-cycles in Sn. Suppose that Γ n k,m be the Cayley graph on subgroup of Sn generated by all permutations in Ωk,m. We find a shortest path joining identity and any vertex of Γ n k,m, for arbitrary natural number k, and m = 2, 3, 4. Also, we calculate the diameter of these Cayley graphs. As an application, we present an algorithm for finding a short expression of a permutation as products of given permutations.
在本文中，我们扩展了B. suceavei和R. strong [Amer]的结果。数学。每月，110(2003)162-162]，他们计算了生成偶数排列所需的最小3循环数。设Ωk,m为c1c2···ck形式的所有排列的集合，其中ci是Sn中的任意m环。设Γ n k,m为Ωk,m中所有排列生成的Sn子群上的Cayley图。对于任意自然数k,m = 2,3,4，我们找到一条最短路径连接单位单位和任意顶点Γ n k,m。同时，我们计算这些Cayley图的直径。作为一个应用，我们提出了一种算法来寻找一个排列的短表达式作为给定排列的乘积。
{"title":"Distance in cayley graphs on permutation groups generated by $k$ $m$-Cycles","authors":"Z. Mostaghim, Mohammad Hossein Ghaffari","doi":"10.22108/TOC.2017.21473","DOIUrl":"https://doi.org/10.22108/TOC.2017.21473","url":null,"abstract":"In this paper, we extend upon the results of B. Suceavă and R. Stong [Amer. Math. Monthly, 110 (2003) 162–162], which they computed the minimum number of 3-cycles needed to generate an even permutation. Let Ωk,m be the set of all permutations of the form c1c2 · · · ck where ci’s are arbitrary m-cycles in Sn. Suppose that Γ n k,m be the Cayley graph on subgroup of Sn generated by all permutations in Ωk,m. We find a shortest path joining identity and any vertex of Γ n k,m, for arbitrary natural number k, and m = 2, 3, 4. Also, we calculate the diameter of these Cayley graphs. As an application, we present an algorithm for finding a short expression of a permutation as products of given permutations.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"6 1","pages":"45-59"},"PeriodicalIF":0.4,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47117788","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}
In this paper we introduce the concept of generalized trees and compute the Hilbert series of their binomial edge ideals.
本文引入了广义树的概念，并计算了其二项边理想的Hilbert级数。
{"title":"On the hilbert series of binomial edge ideals of generalized trees","authors":"M. Saeedi, F. Rahmati","doi":"10.22108/TOC.2017.21463","DOIUrl":"https://doi.org/10.22108/TOC.2017.21463","url":null,"abstract":"In this paper we introduce the concept of generalized trees and compute the Hilbert series of their binomial edge ideals.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"6 1","pages":"11-18"},"PeriodicalIF":0.4,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45458593","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 $delta (G)$, $Delta (G)$ and $gamma(G)$ be the minimum degree, maximum degree and domination number of a graph $G=(V(G), E(G))$, respectively. A partition of $V(G)$, all of whose classes are dominating sets in $G$, is called a domatic partition of $G$. The maximum number of classes of a domatic partition of $G$ is called the domatic number of $G$, denoted $d(G)$. It is well known that $d(G) leq delta(G) + 1$, $d(G)gamma(G) leq |V(G)|$ cite{ch}, and $|V(G)| leq (Delta(G)+1)gamma(G)$ cite{berge}. In this paper, we investigate the graphs $G$ for which all the above inequalities become simultaneously equalities.
{"title":"Common extremal graphs for three inequalities involving domination parameters","authors":"V. Samodivkin","doi":"10.22108/TOC.2017.21464","DOIUrl":"https://doi.org/10.22108/TOC.2017.21464","url":null,"abstract":"Let $delta (G)$, $Delta (G)$ and $gamma(G)$ be the minimum degree, maximum degree and domination number of a graph $G=(V(G), E(G))$, respectively. A partition of $V(G)$, all of whose classes are dominating sets in $G$, is called a domatic partition of $G$. The maximum number of classes of a domatic partition of $G$ is called the domatic number of $G$, denoted $d(G)$. It is well known that $d(G) leq delta(G) + 1$, $d(G)gamma(G) leq |V(G)|$ cite{ch}, and $|V(G)| leq (Delta(G)+1)gamma(G)$ cite{berge}. In this paper, we investigate the graphs $G$ for which all the above inequalities become simultaneously equalities.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"6 1","pages":"1-9"},"PeriodicalIF":0.4,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42042136","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}
Binary array pairs with optimal/ideal correlation values and their algebraic counterparts textquotedblleft difference set pairstextquotedblright;(DSPs) in abelian groups are studied. In addition to generalizing known 1-dimensional (sequences) examples, we provide four new recursive constructions, unifying previously obtained ones. Any further advancements in the construction of binary sequences/arrays with optimal/ideal correlation values (equivalently cyclic/abelian difference sets) would give rise to richer classes of DSPs (and hence binary perfect array pairs). Discrete signals arising from DSPs find applications in cryptography, CDMA systems, radar and wireless communications.
{"title":"Binary sequence/array pairs via diference set pairs: A recursive approach","authors":"K. Arasu, Anika Goyal, Abhishek Puri","doi":"10.22108/TOC.2017.21466","DOIUrl":"https://doi.org/10.22108/TOC.2017.21466","url":null,"abstract":"Binary array pairs with optimal/ideal correlation values and their algebraic counterparts textquotedblleft difference set pairstextquotedblright;(DSPs) in abelian groups are studied. In addition to generalizing known 1-dimensional (sequences) examples, we provide four new recursive constructions, unifying previously obtained ones. Any further advancements in the construction of binary sequences/arrays with optimal/ideal correlation values (equivalently cyclic/abelian difference sets) would give rise to richer classes of DSPs (and hence binary perfect array pairs). Discrete signals arising from DSPs find applications in cryptography, CDMA systems, radar and wireless communications.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"6 1","pages":"19-36"},"PeriodicalIF":0.4,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43944944","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 : 2017-07-01DOI: 10.22108/TOC.2019.105482.1509
P. Sharifani, M. R. Hooshmandasl
For a graph $G=(V,E)$, a set $S subseteq V$ is a $[1,2]$-set if it is a dominating set for $G$ and each vertex $v in V setminus S$ is dominated by at most two vertices of $S$, i.e. $1 leq vert N(v) cap S vert leq 2$. Moreover a set $S subseteq V$ is a total $[1,2]$-set if for each vertex of $V$, it is the case that $1 leq vert N(v) cap S vert leq 2$. The $[1,2]$-domination number of $G$, denoted $gamma_{[1,2]}(G)$, is the minimum number of vertices in a $[1,2]$-set. Every $[1,2]$-set with cardinality of $gamma_{[1,2]}(G)$ is called a $gamma_{[1,2]}$-set. Total $[1,2]$-domination number and $gamma_{t[1,2]}$-sets of $G$ are defined in a similar way. This paper presents a linear time algorithm to find a $gamma_{[1,2]}$-set and a $gamma_{t[1,2]}$-set in generalized series-parallel graphs.
对于图$G=(V,E)$，如果集合$S subseteq V$是$G$的支配集，并且V set- S$中的每个顶点$ V最多被$S$的两个顶点支配，即$1 leq vert N(V) cap S vert leq 2$，则该集合$S subseteq V$是$[1,2]$-set。此外，集合$S subseteq V$是一个总$[1,2]$-如果对于$V$的每个顶点，则$1 leq vert N(V) cap S vert leq 2$。$G$的$[1,2]$支配数，表示$gamma_{[1,2]}(G)$，是$[1,2]$-集合中的最小顶点数。每一个基数为$gamma_{[1,2]}(G)$的$[1,2]$-集称为$gamma_{[1,2]}$-集。Total $[1,2]$-domination number和$gamma_{t[1,2]}$-set of $G$以类似的方式定义。本文给出了一种求广义序列-并行图中$gamma_{[1,2]}$-集和$gamma_{t[1,2]}$-集的线性时间算法。
{"title":"A Linear Algorithm for Computing $gamma_{[1,2]}$-set in Generalized Series-Parallel Graphs","authors":"P. Sharifani, M. R. Hooshmandasl","doi":"10.22108/TOC.2019.105482.1509","DOIUrl":"https://doi.org/10.22108/TOC.2019.105482.1509","url":null,"abstract":"For a graph $G=(V,E)$, a set $S subseteq V$ is a $[1,2]$-set if it is a dominating set for $G$ and each vertex $v in V setminus S$ is dominated by at most two vertices of $S$, i.e. $1 leq vert N(v) cap S vert leq 2$. Moreover a set $S subseteq V$ is a total $[1,2]$-set if for each vertex of $V$, it is the case that $1 leq vert N(v) cap S vert leq 2$. The $[1,2]$-domination number of $G$, denoted $gamma_{[1,2]}(G)$, is the minimum number of vertices in a $[1,2]$-set. Every $[1,2]$-set with cardinality of $gamma_{[1,2]}(G)$ is called a $gamma_{[1,2]}$-set. Total $[1,2]$-domination number and $gamma_{t[1,2]}$-sets of $G$ are defined in a similar way. This paper presents a linear time algorithm to find a $gamma_{[1,2]}$-set and a $gamma_{t[1,2]}$-set in generalized series-parallel graphs.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"18 1","pages":"1-24"},"PeriodicalIF":0.4,"publicationDate":"2017-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78991619","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 emph{bipartite realisation problem} asks for a pair of non-negative, non-increasing integer lists $a:=(a_1,ldots,a_n)$ and $b:=(b_1,ldots,b_{n'})$ if there is a labeled bipartite graph $G(U,V,E)$ (no loops or multiple edges) such that each vertex $u_i in U$ has degree $a_i$ and each vertex $v_i in V$ degree $b_i.$ The Gale-Ryser theorem provides characterisations for the existence of a `realisation' $G(U,V,E)$ that are strongly related to the concept of emph{majorisation}. We prove a generalisation; list pair $(a,b)$ has more realisations than $(a',b),$ if $a'$ majorises $a.$ Furthermore, we give explicitly list pairs which possess the largest number of realisations under all $(a,b)$ with fixed $n$, $n'$ and $m:=sum_{i=1}^n a_i.$ We introduce the notion~emph{minconvex list pairs} for them. If $n$ and $n'$ divide $m,$ minconvex list pairs turn in the special case of two constant lists $a=(frac{m}{n},ldots,frac{m}{n})$ and $b=(frac{m}{n'},ldots,frac{m}{n'}).$
emph{二部实现问题}要求一对非负的，非递增的整数列表$a:=(a_1,ldots,a_n)$和$b:=(b_1,ldots,b_{n'})$，如果有一个标记的二部图$G(U,V,E)$(没有环路或多条边)，使得U$中的每个顶点$u_i都有阶$a_i$和V$ degree $b_i中的每个顶点$v_i。Gale-Ryser定理为“实现”$G(U,V,E)$的存在性提供了特征，这些特征与强调{多数化}的概念密切相关。我们证明了一个概括;列表对$(a,b)$比$(a'，b)有更多的实现，$ if $a'$大写$a。更进一步，我们给出了在所有$(a,b)$具有固定$n$， $n'$和$m的$(a,b)$下具有最大实现数的列表对:=sum_{i=1}^n a_i。我们为它们引入了~emph{最小凸列表对}的概念。如果$n$和$n'$除$m，则$ min凸列表对在两个常量列表的特殊情况下变成$a=(frac{m}{n}，ldots,frac{m}{n})$和$b=(frac{m}{n'}，ldots,frac{m}{n'}) $
{"title":"Majorization and the number of bipartite graphs for given vertex degrees","authors":"A. Berger","doi":"10.22108/TOC.2017.21469","DOIUrl":"https://doi.org/10.22108/TOC.2017.21469","url":null,"abstract":"The emph{bipartite realisation problem} asks for a pair of non-negative, non-increasing integer lists $a:=(a_1,ldots,a_n)$ and $b:=(b_1,ldots,b_{n'})$ if there is a labeled bipartite graph $G(U,V,E)$ (no loops or multiple edges) such that each vertex $u_i in U$ has degree $a_i$ and each vertex $v_i in V$ degree $b_i.$ The Gale-Ryser theorem provides characterisations for the existence of a `realisation' $G(U,V,E)$ that are strongly related to the concept of emph{majorisation}. We prove a generalisation; list pair $(a,b)$ has more realisations than $(a',b),$ if $a'$ majorises $a.$ Furthermore, we give explicitly list pairs which possess the largest number of realisations under all $(a,b)$ with fixed $n$, $n'$ and $m:=sum_{i=1}^n a_i.$ We introduce the notion~emph{minconvex list pairs} for them. If $n$ and $n'$ divide $m,$ minconvex list pairs turn in the special case of two constant lists $a=(frac{m}{n},ldots,frac{m}{n})$ and $b=(frac{m}{n'},ldots,frac{m}{n'}).$","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"60 1","pages":"19-30"},"PeriodicalIF":0.4,"publicationDate":"2017-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68209137","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 $fneq1,3$ be a positive integer. We prove that there exists a numerical semigroup $S$ with embedding dimension three such that $f$ is the Frobenius number of $S$. We also show that the same fact holds for affine semigroups in higher dimensional monoids.
{"title":"On numerical semigroups with embedding dimension three","authors":"A. Mahdavi, F. Rahmati","doi":"10.22108/TOC.2017.20736","DOIUrl":"https://doi.org/10.22108/TOC.2017.20736","url":null,"abstract":"Let $fneq1,3$ be a positive integer. We prove that there exists a numerical semigroup $S$ with embedding dimension three such that $f$ is the Frobenius number of $S$. We also show that the same fact holds for affine semigroups in higher dimensional monoids.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"6 1","pages":"1-6"},"PeriodicalIF":0.4,"publicationDate":"2017-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44754241","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. An edge labeling $f:Eto {0,1}$ induces a vertex labeling $f^+:VtoZ_2$ defined by $f^+(v)equiv sumlimits_{uvin E} f(uv)pmod{2}$ for each $v in V$, where $Z_2={0,1}$ is the additive group of order 2. For $iin{0,1}$, let $e_f(i)=|f^{-1}(i)|$ and $v_f(i)=|(f^+)^{-1}(i)|$. A labeling $f$ is called edge-friendly if $|e_f(1)-e_f(0)|le 1$. $I_f(G)=v_f(1)-v_f(0)$ is called the edge-friendly index of $G$ under an edge-friendly labeling $f$. The full edge-friendly index set of a graph $G$ is the set of all possible edge-friendly indices of $G$. Full edge-friendly index sets of complete bipartite graphs will be determined.
{"title":"Full edge-friendly index sets of complete bipartite graphs","authors":"W. Shiu","doi":"10.22108/TOC.2017.20739","DOIUrl":"https://doi.org/10.22108/TOC.2017.20739","url":null,"abstract":"Let $G=(V,E)$ be a simple graph. An edge labeling $f:Eto {0,1}$ induces a vertex labeling $f^+:VtoZ_2$ defined by $f^+(v)equiv sumlimits_{uvin E} f(uv)pmod{2}$ for each $v in V$, where $Z_2={0,1}$ is the additive group of order 2. For $iin{0,1}$, let $e_f(i)=|f^{-1}(i)|$ and $v_f(i)=|(f^+)^{-1}(i)|$. A labeling $f$ is called edge-friendly if $|e_f(1)-e_f(0)|le 1$. $I_f(G)=v_f(1)-v_f(0)$ is called the edge-friendly index of $G$ under an edge-friendly labeling $f$. The full edge-friendly index set of a graph $G$ is the set of all possible edge-friendly indices of $G$. Full edge-friendly index sets of complete bipartite graphs will be determined.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"6 1","pages":"7-17"},"PeriodicalIF":0.4,"publicationDate":"2017-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42808078","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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