In this paper, we initially introduce the concept of $n$-distance-balanced property which is considered as the generalized concept of distance-balanced property. In our consideration, we also define the new concept locally regularity in order to find a connection between $n$-distance-balanced graphs and their lexicographic product. Furthermore, we include a characteristic method which is practicable and can be used to classify all graphs with $i$-distance-balanced properties for $ i=2,3 $ which is also relevant to the concept of total distance. Moreover, we conclude a connection between distance-balanced and 2-distance-balanced graphs.
{"title":"ON THE NEW EXTENSION OF DISTANCE-BALANCED GRAPHS","authors":"M. Faghani, E. Pourhadi, H. Kharazi","doi":"10.22108/TOC.2016.15048","DOIUrl":"https://doi.org/10.22108/TOC.2016.15048","url":null,"abstract":"In this paper, we initially introduce the concept of $n$-distance-balanced property which is considered as the generalized concept of distance-balanced property. In our consideration, we also define the new concept locally regularity in order to find a connection between $n$-distance-balanced graphs and their lexicographic product. Furthermore, we include a characteristic method which is practicable and can be used to classify all graphs with $i$-distance-balanced properties for $ i=2,3 $ which is also relevant to the concept of total distance. Moreover, we conclude a connection between distance-balanced and 2-distance-balanced graphs.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"5 1","pages":"21-34"},"PeriodicalIF":0.4,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68208507","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 first Zagreb index, $M_1(G)$, and second Zagreb index, $M_2(G)$, of the graph $G$ is defined as $M_{1}(G)=sum_{vin V(G)}d^{2}(v)$ and $M_{2}(G)=sum_{e=uvin E(G)}d(u)d(v),$ where $d(u)$ denotes the degree of vertex $u$. In this paper, the first and second maximum values of the first and second Zagreb indices in the class of all $n-$vertex tetracyclic graphs are presented.
第一萨格勒布指数1 (G),美元和第二萨格勒布指数M_2 (G),美元图G的定义是美元的美元M_ {1} (G) = sum_ {vinV (G)} d ^ {2} (V)和M_美元{2}(G) = sum_ {e = uvin e (G)} (u) d (V),美元在d (u)美元表示顶点u美元的程度。本文给出了所有$n-$顶点四环图的第一类和第二类萨格勒布指数的第一类最大值和第二类最大值。
{"title":"Extremal tetracyclic graphs with respect to the first and second Zagreb indices","authors":"N. Habibi, Tayebeh Dehghan Zadeh, A. Ashrafi","doi":"10.22108/TOC.2016.12878","DOIUrl":"https://doi.org/10.22108/TOC.2016.12878","url":null,"abstract":"The first Zagreb index, $M_1(G)$, and second Zagreb index, $M_2(G)$, of the graph $G$ is defined as $M_{1}(G)=sum_{vin V(G)}d^{2}(v)$ and $M_{2}(G)=sum_{e=uvin E(G)}d(u)d(v),$ where $d(u)$ denotes the degree of vertex $u$. In this paper, the first and second maximum values of the first and second Zagreb indices in the class of all $n-$vertex tetracyclic graphs are presented.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"5 1","pages":"35-55"},"PeriodicalIF":0.4,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68208359","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 with unity. The comaximal ideal graph of $R$, denoted by $mathcal{C}(R)$, is a graph whose vertices are the proper ideals of $R$ which are not contained in the Jacobson radical of $R$, and two vertices $I_1$ and $I_2$ are adjacent if and only if $I_1 +I_2 = R$. In this paper, we classify all comaximal ideal graphs with finite independence number and present a formula to calculate this number. Also, the domination number of $mathcal{C}(R)$ for a ring $R$ is determined. In the last section, we introduce all planar and toroidal comaximal ideal graphs. Moreover, the commutative rings with isomorphic comaximal ideal graphs are characterized. In particular we show that every finite comaximal ideal graph is isomorphic to some $mathcal{C}(mathbb{Z}_n)$.
{"title":"Some results on the comaximal ideal graph of a commutative ring","authors":"H. Dorbidi, R. Manaviyat","doi":"10.22108/TOC.2016.15047","DOIUrl":"https://doi.org/10.22108/TOC.2016.15047","url":null,"abstract":"Let $R$ be a commutative ring with unity. The comaximal ideal graph of $R$, denoted by $mathcal{C}(R)$, is a graph whose vertices are the proper ideals of $R$ which are not contained in the Jacobson radical of $R$, and two vertices $I_1$ and $I_2$ are adjacent if and only if $I_1 +I_2 = R$. In this paper, we classify all comaximal ideal graphs with finite independence number and present a formula to calculate this number. Also, the domination number of $mathcal{C}(R)$ for a ring $R$ is determined. In the last section, we introduce all planar and toroidal comaximal ideal graphs. Moreover, the commutative rings with isomorphic comaximal ideal graphs are characterized. In particular we show that every finite comaximal ideal graph is isomorphic to some $mathcal{C}(mathbb{Z}_n)$.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"5 1","pages":"9-20"},"PeriodicalIF":0.4,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68208332","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 construct a new class of integral bipartite graphs (not necessarily trees) with large even diameters. In fact, for every finite set A of positive integers of size k we construct an integral bipartite graph G of diameter 2 k such that the set of positive eigenvalues of G is exactly A . This class of integral bipartite graphs has never found before.
{"title":"New class of integral bipartite graphs with large diameter","authors":"A. F. Laali, H. Javadi","doi":"10.22108/toc.2016.20738","DOIUrl":"https://doi.org/10.22108/toc.2016.20738","url":null,"abstract":". In this paper, we construct a new class of integral bipartite graphs (not necessarily trees) with large even diameters. In fact, for every finite set A of positive integers of size k we construct an integral bipartite graph G of diameter 2 k such that the set of positive eigenvalues of G is exactly A . This class of integral bipartite graphs has never found before.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"7 1","pages":"13-17"},"PeriodicalIF":0.4,"publicationDate":"2016-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68208571","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 $f^+:Vto Z_2$ 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 $f$. Extreme values of edge-friendly index of complete bipartite graphs will be determined.
{"title":"Extreme edge-friendly indices of complete bipartite graphs","authors":"W. Shiu","doi":"10.22108/TOC.2016.12473","DOIUrl":"https://doi.org/10.22108/TOC.2016.12473","url":null,"abstract":"Let $G=(V,E)$ be a simple graph. An edge labeling $f:Eto {0,1}$ induces a vertex labeling $f^+:Vto Z_2$ 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 $f$. Extreme values of edge-friendly index of complete bipartite graphs will be determined.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"5 1","pages":"11-21"},"PeriodicalIF":0.4,"publicationDate":"2016-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68208281","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 $G$ be a finite simple graph on the vertex set $V(G)$ and let $S subseteq V(G)$. Adding a whisker to $G$ at $x$ means adding a new vertex $y$ and edge $xy$ to $G$ where $x in V(G)$. The graph $Gcup W(S)$ is obtained from $G$ by adding a whisker to every vertex of $S$. We prove that if $Gsetminus S$ is either a graph with no chordless cycle of length other than $3$ or $5$, chordal graph or $C_5$, then $G cup W(S)$ is a vertex decomposable graph.
设$G$是顶点集$V(G)$上的一个有限简单图，设$S subseteq V(G)$。在$x$处向$G$添加须意味着在$G$中添加一个新的顶点$y$和边缘$xy$，其中$x在V(G)$中。图$Gcup W(S)$由$G$通过在$S$的每个顶点上添加晶须而得到。证明了如果$ gset- S$是一个除了$3$或$5$以外没有无弦循环长度的图，有弦图或$C_5$，则$G cup W(S)$是一个顶点可分解图。
{"title":"A NEW CONSTRUCTION FOR VERTEX DECOMPOSABLE GRAPHS","authors":"N. Hajisharifi, A. Tehranian","doi":"10.22108/TOC.2016.13316","DOIUrl":"https://doi.org/10.22108/TOC.2016.13316","url":null,"abstract":"Let $G$ be a finite simple graph on the vertex set $V(G)$ and let $S subseteq V(G)$. Adding a whisker to $G$ at $x$ means adding a new vertex $y$ and edge $xy$ to $G$ where $x in V(G)$. The graph $Gcup W(S)$ is obtained from $G$ by adding a whisker to every vertex of $S$. We prove that if $Gsetminus S$ is either a graph with no chordless cycle of length other than $3$ or $5$, chordal graph or $C_5$, then $G cup W(S)$ is a vertex decomposable graph.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"5 1","pages":"33-38"},"PeriodicalIF":0.4,"publicationDate":"2016-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68208402","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 Wiener index $W(G)$ of a connected graph $G$ is defined as $W(G)=sum_{u,vin V(G)}d_G(u,v)$ where $d_G(u,v)$ is the distance between the vertices $u$ and $v$ of $G$. For $Ssubseteq V(G)$, the Steiner distance $d(S)$ of the vertices of $S$ is the minimum size of a connected subgraph of $G$ whose vertex set is $S$. The $k$-th Steiner Wiener index $SW_k(G)$ of $G$ is defined as $SW_k(G)=sum_{overset{Ssubseteq V(G)}{|S|=k}} d(S)$. We establish expressions for the $k$-th Steiner Wiener index on the join, corona, cluster, lexicographical product, and Cartesian product of graphs.
连通图$G$的Wiener指数$W(G)$定义为$W(G)=sum_{u,vin V(G)}d_G(u, V)$ $其中$d_G(u, V)$是$G$ $ $的顶点$u$和$ V $之间的距离。对于$Ssubseteq V(G)$ $， $S$的顶点的Steiner距离$d(S)$是$ $G$的连通子图的最小大小，其顶点集为$S$ $。k th Steiner维纳美元指数SW_k美元(G) G被定义为美元的美元SW_k美元(G) = sum_{打翻{Ssubseteq V (G)} {S | | = k}} d (S)美元。我们建立了图的连接、冕、聚类、词典积和笛卡尔积上的第k阶Steiner Wiener索引的表达式。
{"title":"Steiner Wiener index of graph products","authors":"Yaoping Mao, Zhao Wang, I. Gutman","doi":"10.22108/TOC.2016.13499","DOIUrl":"https://doi.org/10.22108/TOC.2016.13499","url":null,"abstract":"The Wiener index $W(G)$ of a connected graph $G$ is defined as $W(G)=sum_{u,vin V(G)}d_G(u,v)$ where $d_G(u,v)$ is the distance between the vertices $u$ and $v$ of $G$. For $Ssubseteq V(G)$, the Steiner distance $d(S)$ of the vertices of $S$ is the minimum size of a connected subgraph of $G$ whose vertex set is $S$. The $k$-th Steiner Wiener index $SW_k(G)$ of $G$ is defined as $SW_k(G)=sum_{overset{Ssubseteq V(G)}{|S|=k}} d(S)$. We establish expressions for the $k$-th Steiner Wiener index on the join, corona, cluster, lexicographical product, and Cartesian product of graphs.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"5 1","pages":"39-50"},"PeriodicalIF":0.4,"publicationDate":"2016-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68208453","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 give an algorithm, called T$^{*}$, for finding the k shortest simple paths connecting a certain pair of nodes, $s$ and $t$, in a acyclic digraph. First the nodes of the graph are labeled according to the topological ordering. Then for node $i$ an ordered list of simple $s-i$ paths is created. The length of the list is at most $k$ and it is created by using tournament trees. We prove the correctness of T$^{*}$ and show that its worst-case complexity is $O(m+k n log overline{d})$ in which n is the number of nodes and m is the number of arcs and $overline{d}$ is the mean degree of the graph. The algorithm has a space complexity of $O(kn)$ which entails an important improvement in space complexity. An experimental evaluation of T$^{*}$ is presented which confirms the advantage of our algorithm compared to the most efficient $k$ shortest paths algorithms known so far.
我们给出了一个算法，称为T ^{*}$，用于在无环有向图中寻找连接某一对节点，$s$和$ T$ $的k条最短简单路径。首先，根据拓扑顺序对图中的节点进行标记。然后为节点$i$创建一个简单$s-i$路径的有序列表。列表的长度最多为$k$，它是通过使用锦标赛树创建的。我们证明了T$^{*}$的正确性，并证明了它的最坏情况复杂度为$O(m+k n log overline{d})$，其中n为节点数，m为弧数，$overline{d}$为图的平均度。该算法的空间复杂度为0 (kn)，极大地提高了空间复杂度。提出了T$^{*}$的实验评估，证实了我们的算法与迄今为止已知的最有效的$k$最短路径算法相比的优势。
{"title":"A new $O(m+k n log overline{d})$ algorithm to find the $k$ shortest paths in acyclic digraphs","authors":"M. Kadivar","doi":"10.22108/TOC.2016.12602","DOIUrl":"https://doi.org/10.22108/TOC.2016.12602","url":null,"abstract":"We give an algorithm, called T$^{*}$, for finding the k shortest simple paths connecting a certain pair of nodes, $s$ and $t$, in a acyclic digraph. First the nodes of the graph are labeled according to the topological ordering. Then for node $i$ an ordered list of simple $s-i$ paths is created. The length of the list is at most $k$ and it is created by using tournament trees. We prove the correctness of T$^{*}$ and show that its worst-case complexity is $O(m+k n log overline{d})$ in which n is the number of nodes and m is the number of arcs and $overline{d}$ is the mean degree of the graph. The algorithm has a space complexity of $O(kn)$ which entails an important improvement in space complexity. An experimental evaluation of T$^{*}$ is presented which confirms the advantage of our algorithm compared to the most efficient $k$ shortest paths algorithms known so far.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"64 1","pages":"23-31"},"PeriodicalIF":0.4,"publicationDate":"2016-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68208341","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}
D. Mojdeh, A. Sayed-Khalkhali, H. A. Ahangar, Yancai Zhao
A set $S$ of vertices in a graph $G=(V,E)$ is called a total $k$-distance dominating set if every vertex in $V$ is within distance $k$ of a vertex in $S$. A graph $G$ is total $k$-distance domination-critical if $gamma_{t}^{k} (G - x) < gamma_{t}^{k} (G)$ for any vertex $xin V(G)$. In this paper, we investigate some results on total $k$-distance domination-critical of graphs.
{"title":"Total $k$-distance domination critical graphs","authors":"D. Mojdeh, A. Sayed-Khalkhali, H. A. Ahangar, Yancai Zhao","doi":"10.22108/TOC.2016.11972","DOIUrl":"https://doi.org/10.22108/TOC.2016.11972","url":null,"abstract":"A set $S$ of vertices in a graph $G=(V,E)$ is called a total $k$-distance dominating set if every vertex in $V$ is within distance $k$ of a vertex in $S$. A graph $G$ is total $k$-distance domination-critical if $gamma_{t}^{k} (G - x) < gamma_{t}^{k} (G)$ for any vertex $xin V(G)$. In this paper, we investigate some results on total $k$-distance domination-critical of graphs.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"5 1","pages":"1-9"},"PeriodicalIF":0.4,"publicationDate":"2016-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68208265","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 Gutman index and degree distance of a connected graph $G$ are defined as begin{eqnarray*} textrm{Gut}(G)=sum_{{u,v}subseteq V(G)}d(u)d(v)d_G(u,v), end{eqnarray*} and begin{eqnarray*} DD(G)=sum_{{u,v}subseteq V(G)}(d(u)+d(v))d_G(u,v), end{eqnarray*} respectively, where $d(u)$ is the degree of vertex $u$ and $d_G(u,v)$ is the distance between vertices $u$ and $v$. In this paper, through a recurrence equation for the Wiener index, we study the first two moments of the Gutman index and degree distance of increasing trees.
连通图$G$的Gutman索引和度距离定义为:$ begin{eqnarray*}} $ $ textrm{Gut}(G)=sum_{{u,v}subseteq v (G)}d(u)d(v)d_G(u,v)}(d(u)+d(v))和$ $ begin{eqnarray*}和$ $ begin{eqnarray*} _ (G)=sum_{{u,v}subseteq v (G)}(d(u)+d(v))d_G(u,v)}， $ end{eqnarray*}}，其中$ $d(u)$是顶点$u$的度，$d_G(u,v)$是顶点$u$和$v$ $之间的距离。本文通过维纳指数的递推方程，研究了古特曼指数的前两个矩和递增树的度距离。
{"title":"DEGREE DISTANCE AND GUTMAN INDEX OF INCREASING TREES","authors":"R. Kazemi, Leila Khaleghi Meimondari","doi":"10.22108/TOC.2016.9915","DOIUrl":"https://doi.org/10.22108/TOC.2016.9915","url":null,"abstract":"The Gutman index and degree distance of a connected graph $G$ are defined as begin{eqnarray*} textrm{Gut}(G)=sum_{{u,v}subseteq V(G)}d(u)d(v)d_G(u,v), end{eqnarray*} and begin{eqnarray*} DD(G)=sum_{{u,v}subseteq V(G)}(d(u)+d(v))d_G(u,v), end{eqnarray*} respectively, where $d(u)$ is the degree of vertex $u$ and $d_G(u,v)$ is the distance between vertices $u$ and $v$. In this paper, through a recurrence equation for the Wiener index, we study the first two moments of the Gutman index and degree distance of increasing trees.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"5 1","pages":"23-31"},"PeriodicalIF":0.4,"publicationDate":"2016-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68209063","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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