Pub Date : 2018-06-01DOI: 10.22108/TOC.2017.104212.1496
Qing Zou
Let $f_n$ denotes the $n$th Fubini number. In this paper, first we give upper and lower bounds for the Fubini numbers $f_n$. Then the log-convexity of the Fubini numbers has been obtained. Furthermore we also give the monotonicity of the sequence ${sqrt[n]{f_n}}_{nge 1}$ by using the aforementioned bounds.
{"title":"The log-convexity of the fubini numbers","authors":"Qing Zou","doi":"10.22108/TOC.2017.104212.1496","DOIUrl":"https://doi.org/10.22108/TOC.2017.104212.1496","url":null,"abstract":"Let $f_n$ denotes the $n$th Fubini number. In this paper, first we give upper and lower bounds for the Fubini numbers $f_n$. Then the log-convexity of the Fubini numbers has been obtained. Furthermore we also give the monotonicity of the sequence ${sqrt[n]{f_n}}_{nge 1}$ by using the aforementioned bounds.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"7 1","pages":"17-23"},"PeriodicalIF":0.4,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46650263","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 : 2018-06-01DOI: 10.22108/TOC.2017.102359.1483
T. Mansour, M. Shattuck
In this paper, we consider statistics on permutations of length $n$ represented geometrically as bargraphs having the same number of horizontal steps. More precisely, we find the joint distribution of the descent and up step statistics on the bargraph representations, thereby obtaining a new refined count of permutations of a given length. To do so, we consider the distribution of the parameters on permutations of a more general multiset of which $mathcal{S}_n$ is a subset. In addition to finding an explicit formula for the joint distribution on this multiset, we provide counts for the total number of descents and up steps of all its members, supplying both algebraic and combinatorial proofs. Finally, we derive explicit expressions for the sign balance of these statistics, from which the comparable results on permutations follow as special cases.
{"title":"Combinatorial parameters on bargraphs of permutations","authors":"T. Mansour, M. Shattuck","doi":"10.22108/TOC.2017.102359.1483","DOIUrl":"https://doi.org/10.22108/TOC.2017.102359.1483","url":null,"abstract":"In this paper, we consider statistics on permutations of length $n$ represented geometrically as bargraphs having the same number of horizontal steps. More precisely, we find the joint distribution of the descent and up step statistics on the bargraph representations, thereby obtaining a new refined count of permutations of a given length. To do so, we consider the distribution of the parameters on permutations of a more general multiset of which $mathcal{S}_n$ is a subset. In addition to finding an explicit formula for the joint distribution on this multiset, we provide counts for the total number of descents and up steps of all its members, supplying both algebraic and combinatorial proofs. Finally, we derive explicit expressions for the sign balance of these statistics, from which the comparable results on permutations follow as special cases.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"7 1","pages":"1-16"},"PeriodicalIF":0.4,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48239246","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 : 2018-06-01DOI: 10.22108/TOC.2017.105701.1510
Hamid Damadi, F. Rahmati
{"title":"On matrix and lattice ideals of digraphs","authors":"Hamid Damadi, F. Rahmati","doi":"10.22108/TOC.2017.105701.1510","DOIUrl":"https://doi.org/10.22108/TOC.2017.105701.1510","url":null,"abstract":"","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"7 1","pages":"35-46"},"PeriodicalIF":0.4,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43001934","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 : 2018-06-01DOI: 10.22108/TOC.2017.101076.1462
Meili Liang, Bo Cheng, Jianxi Liu
The harmonic index of a graph G is defined as H(G) = ∑ uv∈E(G) 2 d(u)+d(v) , where d(u) denotes the degree of a vertex u in G. Let G(n, k) be the set of simple n-vertex graphs with minimum degree at least k. In this work we consider the problem of determining the minimum value of the harmonic index and the corresponding extremal graphs among G(n, k). We solve the problem for each integer k(1 ≤ k ≤ n/2) and show the corresponding extremal graph is the complete split graph K∗ k,n−k. This result together with our previous result which solve the problem for each integer k(n/2 ≤ k ≤ n−1) give a complete solution of the problem.
图G的谐波指数被定义为H (G) =∑紫外线∈E (G) 2 d (u) + d (v),在d (u)表示一个顶点的度u G .让G (n, k)是一组简单的n点图以最小程度至少k。这项工作我们考虑的问题确定谐波指标的最小值和相应的极值图G (n, k)。我们解决问题对于每个整数k (k 1≤≤n / 2)并显示相应的极值图是完整的分割图k∗k, n−k。该结果与我们之前求解每整数k(n/2≤k≤n−1)的结果一起给出了问题的完全解。
{"title":"Solution to the minimum harmonic index of graphs with given minimum degree","authors":"Meili Liang, Bo Cheng, Jianxi Liu","doi":"10.22108/TOC.2017.101076.1462","DOIUrl":"https://doi.org/10.22108/TOC.2017.101076.1462","url":null,"abstract":"The harmonic index of a graph G is defined as H(G) = ∑ uv∈E(G) 2 d(u)+d(v) , where d(u) denotes the degree of a vertex u in G. Let G(n, k) be the set of simple n-vertex graphs with minimum degree at least k. In this work we consider the problem of determining the minimum value of the harmonic index and the corresponding extremal graphs among G(n, k). We solve the problem for each integer k(1 ≤ k ≤ n/2) and show the corresponding extremal graph is the complete split graph K∗ k,n−k. This result together with our previous result which solve the problem for each integer k(n/2 ≤ k ≤ n−1) give a complete solution of the problem.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"7 1","pages":"25-33"},"PeriodicalIF":0.4,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43248343","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 $R$ be a commutative ring and $M$ an $R$-module. In this article, we introduce a new generalization of the annihilating-ideal graph of commutative rings to modules. The annihilating submodule graph of $M$, denoted by $Bbb G(M)$, is an undirected graph with vertex set $Bbb A^*(M)$ and two distinct elements $N$ and $K$ of $Bbb A^*(M)$ are adjacent if $N*K=0$. In this paper we show that $Bbb G(M)$ is a connected graph, ${rm diam}(Bbb G(M))leq 3$, and ${rm gr}(Bbb G(M))leq 4$ if $Bbb G(M)$ contains a cycle. Moreover, $Bbb G(M)$ is an empty graph if and only if ${rm ann}(M)$ is a prime ideal of $R$ and $Bbb A^*(M)neq Bbb S(M)setminus {0}$ if and only if $M$ is a uniform $R$-module, ${rm ann}(M)$ is a semi-prime ideal of $R$ and $Bbb A^*(M)neq Bbb S(M)setminus {0}$. Furthermore, $R$ is a field if and only if $Bbb G(M)$ is a complete graph, for every $Min R-{rm Mod}$. If $R$ is a domain, for every divisible module $Min R-{rm Mod}$, $Bbb G(M)$ is a complete graph with $Bbb A^*(M)=Bbb S(M)setminus {0}$. Among other things, the properties of a reduced $R$-module $M$ are investigated when $Bbb G(M)$ is a bipartite graph.
{"title":"Annihilating submodule graph for modules","authors":"S. Safaeeyan","doi":"10.22108/toc.2017.21462","DOIUrl":"https://doi.org/10.22108/toc.2017.21462","url":null,"abstract":"Let $R$ be a commutative ring and $M$ an $R$-module. In this article, we introduce a new generalization of the annihilating-ideal graph of commutative rings to modules. The annihilating submodule graph of $M$, denoted by $Bbb G(M)$, is an undirected graph with vertex set $Bbb A^*(M)$ and two distinct elements $N$ and $K$ of $Bbb A^*(M)$ are adjacent if $N*K=0$. In this paper we show that $Bbb G(M)$ is a connected graph, ${rm diam}(Bbb G(M))leq 3$, and ${rm gr}(Bbb G(M))leq 4$ if $Bbb G(M)$ contains a cycle. Moreover, $Bbb G(M)$ is an empty graph if and only if ${rm ann}(M)$ is a prime ideal of $R$ and $Bbb A^*(M)neq Bbb S(M)setminus {0}$ if and only if $M$ is a uniform $R$-module, ${rm ann}(M)$ is a semi-prime ideal of $R$ and $Bbb A^*(M)neq Bbb S(M)setminus {0}$. Furthermore, $R$ is a field if and only if $Bbb G(M)$ is a complete graph, for every $Min R-{rm Mod}$. If $R$ is a domain, for every divisible module $Min R-{rm Mod}$, $Bbb G(M)$ is a complete graph with $Bbb A^*(M)=Bbb S(M)setminus {0}$. Among other things, the properties of a reduced $R$-module $M$ are investigated when $Bbb G(M)$ is a bipartite graph.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"7 1","pages":"1-12"},"PeriodicalIF":0.4,"publicationDate":"2018-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47546622","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}
For any prime $p$ let $C_p(G)$ be the $p$-ary code spanned by the rows of the incidence matrix $G$ of a graph $Gamma$. Let $Gamma$ be the incidence graph of a flag-transitive symmetric design $D$. We show that any flag-transitive automorphism group of $D$ can be used as a PD-set for full error correction for the linear code $C_p(G)$ (with any information set). It follows that such codes derived from flag-transitive symmetric designs can be decoded using permutation decoding. In that way to each flag-transitive symmetric $(v, k, lambda)$ design we associate a linear code of length $vk$ that is permutation decodable. PD-sets obtained in the described way are usually of large cardinality. By studying codes arising from some flag-transitive symmetric designs we show that smaller PD-sets can be found for specific information sets.
{"title":"PD-sets for codes related to flag-transitive symmetric designs","authors":"D. Crnković, Nina Mostarac","doi":"10.22108/TOC.2017.21615","DOIUrl":"https://doi.org/10.22108/TOC.2017.21615","url":null,"abstract":"For any prime $p$ let $C_p(G)$ be the $p$-ary code spanned by the rows of the incidence matrix $G$ of a graph $Gamma$. Let $Gamma$ be the incidence graph of a flag-transitive symmetric design $D$. We show that any flag-transitive automorphism group of $D$ can be used as a PD-set for full error correction for the linear code $C_p(G)$ (with any information set). It follows that such codes derived from flag-transitive symmetric designs can be decoded using permutation decoding. In that way to each flag-transitive symmetric $(v, k, lambda)$ design we associate a linear code of length $vk$ that is permutation decodable. PD-sets obtained in the described way are usually of large cardinality. By studying codes arising from some flag-transitive symmetric designs we show that smaller PD-sets can be found for specific information sets.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"7 1","pages":"37-50"},"PeriodicalIF":0.4,"publicationDate":"2018-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43763854","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 obtain α as coefficient for the G = Kαn∪K(1−α)n and by which we discuss Nikiforov’s conjecture for λ1 and Aouchiche and Hansen’s conjecture for q1 in Nordhaus-Gaddum type inequalities. Furthermore, by the properties of the products of graphs we put forward a new approach to find some bounds of Nordhaus-Gaddum type inequalities.
{"title":"Products of graphs and Nordhaus-Gaddum type inequalities for eigenvalues","authors":"Nastaran Keyvan, F. Rahmati","doi":"10.22108/TOC.2017.21474","DOIUrl":"https://doi.org/10.22108/TOC.2017.21474","url":null,"abstract":"In this paper, we obtain α as coefficient for the G = Kαn∪K(1−α)n and by which we discuss Nikiforov’s conjecture for λ1 and Aouchiche and Hansen’s conjecture for q1 in Nordhaus-Gaddum type inequalities. Furthermore, by the properties of the products of graphs we put forward a new approach to find some bounds of Nordhaus-Gaddum type inequalities.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"7 1","pages":"31-36"},"PeriodicalIF":0.4,"publicationDate":"2018-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45946469","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 harmonic index of a graph $G$ is defined as the sum of the weights $frac{2}{deg_G(u)+deg_G(v)}$ of all edges $uv$ of $G$, where $deg_G(u)$ denotes the degree of a vertex $u$ in $G$. In this paper, we study the harmonic index of subdivision graphs, $t$-subdivision graphs and also, $S$-sum and $S_t$-sum of graphs.
{"title":"The harmonic index of subdivision graphs","authors":"B. N. Onagh","doi":"10.22108/TOC.2017.21471","DOIUrl":"https://doi.org/10.22108/TOC.2017.21471","url":null,"abstract":"The harmonic index of a graph $G$ is defined as the sum of the weights $frac{2}{deg_G(u)+deg_G(v)}$ of all edges $uv$ of $G$, where $deg_G(u)$ denotes the degree of a vertex $u$ in $G$. In this paper, we study the harmonic index of subdivision graphs, $t$-subdivision graphs and also, $S$-sum and $S_t$-sum of graphs.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"6 1","pages":"15-27"},"PeriodicalIF":0.4,"publicationDate":"2017-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41850355","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}
H. Deng, S. Balachandran, S. Ayyaswamy, Y. B. Venkatakrishnan
The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity $eccleft(Gright)$ of a graph $G$ is the mean value of eccentricities of all vertices of $G$. The harmonic index $Hleft(Gright)$ of a graph $G$ is defined as the sum of $frac{2}{d_{i}+d_{j}}$ over all edges $v_{i}v_{j}$ of $G$, where $d_{i}$ denotes the degree of a vertex $v_{i}$ in $G$. In this paper, we determine the unique tree with minimum average eccentricity among the set of trees with given number of pendent vertices and determine the unique tree with maximum average eccentricity among the set of $n$-vertex trees with two adjacent vertices of maximum degree $Delta$, where $ngeq 2Delta$. Also, we give some relations between the average eccentricity, the harmonic index and the largest signless Laplacian eigenvalue, and strengthen a result on the Randi'{c} index and the largest signless Laplacian eigenvalue conjectured by Hansen and Lucas cite{hl}.
{"title":"On the average eccentricity, the harmonic index and the largest signless Laplacian eigenvalue of a graph","authors":"H. Deng, S. Balachandran, S. Ayyaswamy, Y. B. Venkatakrishnan","doi":"10.22108/TOC.2017.21470","DOIUrl":"https://doi.org/10.22108/TOC.2017.21470","url":null,"abstract":"The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity $eccleft(Gright)$ of a graph $G$ is the mean value of eccentricities of all vertices of $G$. The harmonic index $Hleft(Gright)$ of a graph $G$ is defined as the sum of $frac{2}{d_{i}+d_{j}}$ over all edges $v_{i}v_{j}$ of $G$, where $d_{i}$ denotes the degree of a vertex $v_{i}$ in $G$. In this paper, we determine the unique tree with minimum average eccentricity among the set of trees with given number of pendent vertices and determine the unique tree with maximum average eccentricity among the set of $n$-vertex trees with two adjacent vertices of maximum degree $Delta$, where $ngeq 2Delta$. Also, we give some relations between the average eccentricity, the harmonic index and the largest signless Laplacian eigenvalue, and strengthen a result on the Randi'{c} index and the largest signless Laplacian eigenvalue conjectured by Hansen and Lucas cite{hl}.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"6 1","pages":"43-50"},"PeriodicalIF":0.4,"publicationDate":"2017-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47442967","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 regular graph of ideals of the commutative ring $R$, denoted by ${Gamma_{reg}}(R)$, is a graph whose vertex set is the set of all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if either $I$ contains a $J$-regular element or $J$ contains an $I$-regular element. In this paper, it is proved that the radius of $Gamma_{reg}(R)$ equals $3$. The central vertices of $Gamma_{reg}(R)$ are determined, too.
{"title":"The central vertices and radius of the regular graph of ideals","authors":"F. Shaveisi","doi":"10.22108/TOC.2017.21472","DOIUrl":"https://doi.org/10.22108/TOC.2017.21472","url":null,"abstract":"The regular graph of ideals of the commutative ring $R$, denoted by ${Gamma_{reg}}(R)$, is a graph whose vertex set is the set of all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if either $I$ contains a $J$-regular element or $J$ contains an $I$-regular element. In this paper, it is proved that the radius of $Gamma_{reg}(R)$ equals $3$. The central vertices of $Gamma_{reg}(R)$ are determined, too.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"6 1","pages":"1-13"},"PeriodicalIF":0.4,"publicationDate":"2017-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47284918","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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