Pub Date : 2020-03-01DOI: 10.22108/TOC.2019.119856.1682
S. Yassemi
We define a refinement of the notion of Leray simplicial complexes and study its properties. Moreover, we translate some of our results to the language of commutative algebra.
{"title":"On a generalization of Leray simplicial complexes","authors":"S. Yassemi","doi":"10.22108/TOC.2019.119856.1682","DOIUrl":"https://doi.org/10.22108/TOC.2019.119856.1682","url":null,"abstract":"We define a refinement of the notion of Leray simplicial complexes and study its properties. Moreover, we translate some of our results to the language of commutative algebra.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"9 1","pages":"25-30"},"PeriodicalIF":0.4,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44703760","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 : 2020-03-01DOI: 10.22108/TOC.2020.116255.1630
Andrea Švob
In this paper we classify distance-regular graphs, including strongly regular graphs, admitting a transitive action of the linear groups $L(3,2)$, $L(3,3)$, $L(3,4)$ and $L(3,5)$ for which the rank of the permutation representation is at most 15. We give details about constructed graphs. In addition, we construct self-orthogonal codes from distance-regular graphs obtained in this paper.
{"title":"Transitive distance-regular graphs from linear groups $L(3,q)$, $q = 2,3,4,5$","authors":"Andrea Švob","doi":"10.22108/TOC.2020.116255.1630","DOIUrl":"https://doi.org/10.22108/TOC.2020.116255.1630","url":null,"abstract":"In this paper we classify distance-regular graphs, including strongly regular graphs, admitting a transitive action of the linear groups $L(3,2)$, $L(3,3)$, $L(3,4)$ and $L(3,5)$ for which the rank of the permutation representation is at most 15. We give details about constructed graphs. In addition, we construct self-orthogonal codes from distance-regular graphs obtained in this paper.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"9 1","pages":"49-60"},"PeriodicalIF":0.4,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43927950","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 : 2020-01-05DOI: 10.22108/TOC.2020.107875.1529
H. Galeana-Sánchez, R. Rojas-Monroy, Maria Del Rocio Sanchez Lopez, Berta Zavala-Santana
Let $H$ be a digraph possibly with loops and $D$ a digraph without loops whose arcs are colored with the vertices of $H$ ($D$ is said to be an $H$-colored digraph). A directed walk $W$ in $D$ is said to be an $H$-walk if and only if the consecutive colors encountered on $W$ form a directed walk in $H$. A subset $N$ of the vertices of $D$ is said to be an $H$-kernel by walks if (1) for every pair of different vertices in $N$ there is no $H$-walk between them ($N$ is $H$-independent by walks) and (2) for each vertex $u$ in $V$($D$)-$N$ there exists an $H$-walk from $u$ to $N$ in $D$ ($N$ is $H$-absorbent by walks). Suppose that $D$ is a digraph possibly infinite. In this paper we will work with the subdivision digraph $S_H$($D$) of $D$, where $S_H$($D$) is an $H$-colored digraph defined as follows: $V$($S_H$($D$)) = $V$($D$) $cup$ $A$($D$) and $A$($S_H$($D$)) = {($u$,$a$) : $a$ = ($u$,$v$) $in$ $A$($D$)} $cup$ {($a$,$v$) : $a$ = ($u$,$v$) $in$ $A$($D$)}, where ($u$, $a$, $v$) is an $H$-walk in $S_H$($D$) for every $a$ = ($u$,$v$) in $A$($D$). We will show sufficient conditions on $D$ and on $S_H$($D$) which guarantee the existence or uniqueness of $H$-kernels by walks in $S_H$($D$).
设$H$是一个可能有圈的有向图，$D$是一个没有圈的有向图，其弧线用$H$的顶点着色($D$被称为$H$着色的有向图)。$D$中的有向行走$W$被称为$H$-行走当且仅当$W$上遇到的连续颜色形成$H$的有向行走。$D$的顶点子集$N$被称为$H$-游动核，如果(1)对于$N$中的每一对不同的顶点，它们之间没有$H$-游动($N$是$H$-游动独立的)，并且(2)对于$V$($D$)中的每个顶点$u$ -$N$存在$H$-游动从$u$到$D$ ($N$是$H$-游动吸收)。假设$D$是有向图，可能是无限的。在本文中,我们将与细分有向图S_H (D)美元美元D,美元,S_H美元($ D $)是一种H的美元有向图定义如下:V美元(S_H (D)美元美元)= V (D)美元美元杯美元美元(D)美元和美元美元(S_H (D)美元美元)= {(u,美元美元美元):$ $ = (u美元,美元V $)在$ $美元(D)美元}$杯${(美元,美元V $):$ $ = (u美元,美元V $)在$ $美元(D)美元},,(u,美元美元美元,V)美元是一个H走美元S_H (D)美元美元每一个美元= (u美元,美元V $)在一个美元($ D $)。我们将给出$D$和$S_H$($D$)上的充分条件，通过$S_H$($D$)的遍历来保证$H$-核的存在或唯一性。
{"title":"$H$-kernels by walks in subdivision digraph","authors":"H. Galeana-Sánchez, R. Rojas-Monroy, Maria Del Rocio Sanchez Lopez, Berta Zavala-Santana","doi":"10.22108/TOC.2020.107875.1529","DOIUrl":"https://doi.org/10.22108/TOC.2020.107875.1529","url":null,"abstract":"Let $H$ be a digraph possibly with loops and $D$ a digraph without loops whose arcs are colored with the vertices of $H$ ($D$ is said to be an $H$-colored digraph). A directed walk $W$ in $D$ is said to be an $H$-walk if and only if the consecutive colors encountered on $W$ form a directed walk in $H$. A subset $N$ of the vertices of $D$ is said to be an $H$-kernel by walks if (1) for every pair of different vertices in $N$ there is no $H$-walk between them ($N$ is $H$-independent by walks) and (2) for each vertex $u$ in $V$($D$)-$N$ there exists an $H$-walk from $u$ to $N$ in $D$ ($N$ is $H$-absorbent by walks). Suppose that $D$ is a digraph possibly infinite. In this paper we will work with the subdivision digraph $S_H$($D$) of $D$, where $S_H$($D$) is an $H$-colored digraph defined as follows: $V$($S_H$($D$)) = $V$($D$) $cup$ $A$($D$) and $A$($S_H$($D$)) = {($u$,$a$) : $a$ = ($u$,$v$) $in$ $A$($D$)} $cup$ {($a$,$v$) : $a$ = ($u$,$v$) $in$ $A$($D$)}, where ($u$, $a$, $v$) is an $H$-walk in $S_H$($D$) for every $a$ = ($u$,$v$) in $A$($D$). We will show sufficient conditions on $D$ and on $S_H$($D$) which guarantee the existence or uniqueness of $H$-kernels by walks in $S_H$($D$).","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"9 1","pages":"61-75"},"PeriodicalIF":0.4,"publicationDate":"2020-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48239239","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 : 2019-12-31DOI: 10.22108/TOC.2019.117529.1651
M. Nikmehr
Let $R$ be a ring and $alpha$ be a ring endomorphism of $R$. The undirected nilpotent graph of $R$, denoted by $Gamma_N(R)$, is a graph with vertex set $Z_N(R)^*$, and two distinct vertices $x$ and $y$ are connected by an edge if and only if $xy$ is nilpotent, where $Z_N(R)={xin R;|; xy; rm{is; nilpotent,;for; some}; yin R^*}.$ In this article, we investigate the interplay between the ring theoretical properties of a skew polynomial ring $R[x;alpha]$ and the graph-theoretical properties of its nilpotent graph $Gamma_N(R[x;alpha])$. It is shown that if $R$ is a symmetric and $alpha$-compatible with exactly two minimal primes, then $diam(Gamma_N(R[x,alpha]))=2$. Also we prove that $Gamma_N(R)$ is a complete graph if and only if $R$ is isomorphic to $Z_2timesZ_2$.
{"title":"Nilpotent graphs of skew polynomial rings over non-commutative rings","authors":"M. Nikmehr","doi":"10.22108/TOC.2019.117529.1651","DOIUrl":"https://doi.org/10.22108/TOC.2019.117529.1651","url":null,"abstract":"Let $R$ be a ring and $alpha$ be a ring endomorphism of $R$. The undirected nilpotent graph of $R$, denoted by $Gamma_N(R)$, is a graph with vertex set $Z_N(R)^*$, and two distinct vertices $x$ and $y$ are connected by an edge if and only if $xy$ is nilpotent, where $Z_N(R)={xin R;|; xy; rm{is; nilpotent,;for; some}; yin R^*}.$ In this article, we investigate the interplay between the ring theoretical properties of a skew polynomial ring $R[x;alpha]$ and the graph-theoretical properties of its nilpotent graph $Gamma_N(R[x;alpha])$. It is shown that if $R$ is a symmetric and $alpha$-compatible with exactly two minimal primes, then $diam(Gamma_N(R[x,alpha]))=2$. Also we prove that $Gamma_N(R)$ is a complete graph if and only if $R$ is isomorphic to $Z_2timesZ_2$.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48218888","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 : 2019-12-01DOI: 10.22108/TOC.2019.116001.1625
M. Azari
The generalized Zagreb index is an extension of both ordinary and variable Zagreb indices. In this paper, we present exact formulae for the values of the generalized Zagreb index for product graphs. Results are applied to some graphs of general and chemical interest such as nanotubes and nanotori.
{"title":"Generalized Zagreb index of product graphs","authors":"M. Azari","doi":"10.22108/TOC.2019.116001.1625","DOIUrl":"https://doi.org/10.22108/TOC.2019.116001.1625","url":null,"abstract":"The generalized Zagreb index is an extension of both ordinary and variable Zagreb indices. In this paper, we present exact formulae for the values of the generalized Zagreb index for product graphs. Results are applied to some graphs of general and chemical interest such as nanotubes and nanotori.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"8 1","pages":"35-48"},"PeriodicalIF":0.4,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42526315","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 : 2019-12-01DOI: 10.22108/TOC.2019.117338.1648
Zahra Kharaghani
We consider some combinatorics of elliptic root systems of type $A_1$. In particular, with respect to a fixed reflectable base, we give a precise description of the positive roots in terms of a ``positivity'' theorem. Also the set of reduced words of the corresponding Weyl group is precisely described. These then lead to a new characterization of the core of the corresponding Lie algebra, namely we show that the core is generated by positive root spaces.
{"title":"Elliptic root systems of type $A_1$, a combinatorial study","authors":"Zahra Kharaghani","doi":"10.22108/TOC.2019.117338.1648","DOIUrl":"https://doi.org/10.22108/TOC.2019.117338.1648","url":null,"abstract":"We consider some combinatorics of elliptic root systems of type $A_1$. In particular, with respect to a fixed reflectable base, we give a precise description of the positive roots in terms of a ``positivity'' theorem. Also the set of reduced words of the corresponding Weyl group is precisely described. These then lead to a new characterization of the core of the corresponding Lie algebra, namely we show that the core is generated by positive root spaces.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"8 1","pages":"11-21"},"PeriodicalIF":0.4,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48038079","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 : 2019-11-10DOI: 10.22108/TOC.2019.114742.1612
Manjit Singh
Let $mathcal{S}_q$ denote the group of all square elements in the multiplicative group $mathbb{F}_q^*$ of a finite field $mathbb{F}_q$ of odd characteristic containing $q$ elements. Let $mathcal{O}_q$ be the set of all odd order elements of $mathbb{F}_q^*$. Then $mathcal{O}_q$ turns up as a subgroup of $mathcal{S}_q$. In this paper, we show that $mathcal{O}_q=langle4rangle$ if $q=2t+1$ and, $mathcal{O}_q=langle trangle $ if $q=4t+1$, where $q$ and $t$ are odd primes. Further, we determine the coefficients of irreducible factors of $x^{2^nt}-1$ using generators of these special subgroups of $mathbb{F}_q^*$
{"title":"Some subgroups of $mathbb{F}_q^*$ and explicit factors of $x^{2^nd}-1 in mathbb{F}_q[x]$","authors":"Manjit Singh","doi":"10.22108/TOC.2019.114742.1612","DOIUrl":"https://doi.org/10.22108/TOC.2019.114742.1612","url":null,"abstract":"Let $mathcal{S}_q$ denote the group of all square elements in the multiplicative group $mathbb{F}_q^*$ of a finite field $mathbb{F}_q$ of odd characteristic containing $q$ elements. Let $mathcal{O}_q$ be the set of all odd order elements of $mathbb{F}_q^*$. Then $mathcal{O}_q$ turns up as a subgroup of $mathcal{S}_q$. In this paper, we show that $mathcal{O}_q=langle4rangle$ if $q=2t+1$ and, $mathcal{O}_q=langle trangle $ if $q=4t+1$, where $q$ and $t$ are odd primes. Further, we determine the coefficients of irreducible factors of $x^{2^nt}-1$ using generators of these special subgroups of $mathbb{F}_q^*$","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"8 1","pages":"23-33"},"PeriodicalIF":0.4,"publicationDate":"2019-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48596918","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 : 2019-10-12DOI: 10.22108/TOC.2019.115674.1622
K. Das, M. Tavakoli
Metric dimension and defensive $k$-alliance number are two distance-based graph invariants which have applications in robot navigation, quantitative analysis of secondary RNA structures, national defense and fault-tolerant computing. In this paper, some bounds for metric dimension and defensive $k$-alliance of deleted lexicographic product of graphs are presented. We also show that the bounds are sharp.
{"title":"Bounds for metric dimension and defensive k-alliance of graphs under deleted lexicographic product","authors":"K. Das, M. Tavakoli","doi":"10.22108/TOC.2019.115674.1622","DOIUrl":"https://doi.org/10.22108/TOC.2019.115674.1622","url":null,"abstract":"Metric dimension and defensive $k$-alliance number are two distance-based graph invariants which have applications in robot navigation, quantitative analysis of secondary RNA structures, national defense and fault-tolerant computing. In this paper, some bounds for metric dimension and defensive $k$-alliance of deleted lexicographic product of graphs are presented. We also show that the bounds are sharp.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"9 1","pages":"31-39"},"PeriodicalIF":0.4,"publicationDate":"2019-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45203334","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 : 2019-09-01DOI: 10.22108/TOC.2019.115147.1615
Majid Aghel, A. Erfanian, A. Ashrafi
Let G be a simple graph. The graph G is called a quasi unicyclic graph if there exists a vertex x ∈ V (G) such that G−x is a connected graph with a unique cycle. Moreover, the first and the second Zagreb indices of G denoted by M1(G) and M2(G), are the sum of deg (u) overall vertices u in G and the sum of deg(u) deg(v) of all edges uv of G, respectively. The first and the second Zagreb indices are defined relative to the degree of vertices. In this paper, sharp upper and lower bounds for the first and the second Zagreb indices of quasi unicyclic graphs are given. 1. Basic Definitions The first and the second Zagreb indices are among the oldest topological indices defined in 1972 by Gutman and Trinajstić [9]. These numbers have been used to study the molecular complexity, chirality and some other chemical quantities. The first Zagreb index is defined as the sum of the squares of the degrees of the vertices, i.e. M1 (G) = ∑ u∈V (G) deg (u) and the second Zagreb index is the sum of deg(u)deg(v) overall edges uv of G. This means that M2 (G) = ∑ uv∈E(G) deg(u)deg(v). The first and the second Zagreb indices are defined relative to the degree of vertices, which we summarize them without referring to the degree of vertices. MSC(2010): Primary: 05C35; Secondary: 05C07.
{"title":"ON THE FIRST AND SECOND ZAGREB INDICES OF QUASI UNICYCLIC GRAPHS","authors":"Majid Aghel, A. Erfanian, A. Ashrafi","doi":"10.22108/TOC.2019.115147.1615","DOIUrl":"https://doi.org/10.22108/TOC.2019.115147.1615","url":null,"abstract":"Let G be a simple graph. The graph G is called a quasi unicyclic graph if there exists a vertex x ∈ V (G) such that G−x is a connected graph with a unique cycle. Moreover, the first and the second Zagreb indices of G denoted by M1(G) and M2(G), are the sum of deg (u) overall vertices u in G and the sum of deg(u) deg(v) of all edges uv of G, respectively. The first and the second Zagreb indices are defined relative to the degree of vertices. In this paper, sharp upper and lower bounds for the first and the second Zagreb indices of quasi unicyclic graphs are given. 1. Basic Definitions The first and the second Zagreb indices are among the oldest topological indices defined in 1972 by Gutman and Trinajstić [9]. These numbers have been used to study the molecular complexity, chirality and some other chemical quantities. The first Zagreb index is defined as the sum of the squares of the degrees of the vertices, i.e. M1 (G) = ∑ u∈V (G) deg (u) and the second Zagreb index is the sum of deg(u)deg(v) overall edges uv of G. This means that M2 (G) = ∑ uv∈E(G) deg(u)deg(v). The first and the second Zagreb indices are defined relative to the degree of vertices, which we summarize them without referring to the degree of vertices. MSC(2010): Primary: 05C35; Secondary: 05C07.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"8 1","pages":"29-39"},"PeriodicalIF":0.4,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43417922","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 : 2019-09-01DOI: 10.22108/TOC.2019.112258.1579
C. Mynhardt, Jane L. Wodlinger
We derive a new sharp lower bound on the $k$-conversion number of graphs of maximum degree $k+1$. This generalizes a result of W.~Staton [Induced forests in cubic graphs, Discrete Math.,49 (1984) 175--178], which established a lower bound on the $k$-conversion number of $(k+1)$-regular graphs.
{"title":"A lower bound on the $k$-conversion number of graphs of maximum degree $k+1$","authors":"C. Mynhardt, Jane L. Wodlinger","doi":"10.22108/TOC.2019.112258.1579","DOIUrl":"https://doi.org/10.22108/TOC.2019.112258.1579","url":null,"abstract":"We derive a new sharp lower bound on the $k$-conversion number of graphs of maximum degree $k+1$. This generalizes a result of W.~Staton [Induced forests in cubic graphs, Discrete Math.,49 (1984) 175--178], which established a lower bound on the $k$-conversion number of $(k+1)$-regular graphs.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":" ","pages":"1-12"},"PeriodicalIF":0.4,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47258155","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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