University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
5
4
2016
12
01
Cacti with extremal PI Index
1
8
EN
Chunxiang
Wang
Central China Normal University
wcxiang@mail.ccnu.edu.cn
Shaohui
Wang
University of Mississippi
shaohuiwang@yahoo.com
Bing
Wei
University of Mississippi
bwei@olemiss.edu
10.22108/toc.2016.14786
The vertex PI index $PI(G) = sum_{xy in E(G)} [n_{xy}(x) + n_{xy}(y)]$ is a distance-based molecular structure descriptor, where $n_{xy}(x)$ denotes the number of vertices which are closer to the vertex $x$ than to the vertex $y$ and which has been the considerable research in computational chemistry dating back to Harold Wiener in 1947. A connected graph is a cactus if any two of its cycles have at most one common vertex. In this paper, we completely determine the extremal graphs with the greatest and smallest vertex PI indices mong all cacti with a fixed number of vertices. As a consequence, we obtain the sharp bounds with corresponding extremal cacti and extend a known result.
Distance,Extremal bounds,PI index,Cacti
http://toc.ui.ac.ir/article_14786.html
http://toc.ui.ac.ir/article_14786_f95e820e8bf0d1325600f95c8a3d7a24.pdf
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
5
4
2016
12
01
Some results on the comaximal ideal graph of a commutative ring
9
20
EN
Hamid Reza
Dorbidi
University of Jiroft,Jiroft, Kerman, Iran
hr_dorbidi@yahoo.com
Raoufeh
Manaviyat
Payame Noor University, Tehran, Iran
r.manaviyat@gmail.com
10.22108/toc.2016.15047
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 $mathcal{C}(mathbb{Z}_n)$.
Comaximal ideal graph,Genus of graph,Domination Number,Independence number
http://toc.ui.ac.ir/article_15047.html
http://toc.ui.ac.ir/article_15047_e2760f540dc55e62152260c257848270.pdf
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
5
4
2016
12
01
On the new extension of distance-balanced graphs
21
34
EN
Morteza
Faghani
Chief of PNU Saveh branch
m_faghani@pnu.ac.ir
Ehsan
Pourhadi
Comprehensive Imam Hossein University
epourhadi@iust.ac.ir
Hassan
Kharazi
Comprehensive Imam Hossein University
hkharazi@ihu.ac.ir
10.22108/toc.2016.15048
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.
$n$-distance-balanced property,lexicographic product,total distance
http://toc.ui.ac.ir/article_15048.html
http://toc.ui.ac.ir/article_15048_3968109258ac5aaddf5a16c03fc677d5.pdf
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
5
4
2016
12
01
Extremal tetracyclic graphs with respect to the first and second Zagreb indices
35
55
EN
Nader
Habibi
university of Ayatollah Al-ozma
nader.habibi@ymail.com
Tayebeh
Dehghan Zadeh
University of Kashan
ta.dehghanzadeh@gmail.com
Ali Reza
Ashrafi
University of Kashan
ashrafi@kashan.ac.ir
10.22108/toc.2016.12878
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.
First Zagreb index,second Zagreb index,tetracyclic graph
http://toc.ui.ac.ir/article_12878.html
http://toc.ui.ac.ir/article_12878_b7525583ad7d958b2f5cb6c2d9eabfdb.pdf
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
5
4
2016
12
01
Congruences from $q$-Catalan Identities
57
67
EN
Qing
Zou
Department of Mathematics, The University of Iowa
zou-qing@uiowa.edu
10.22108/toc.2016.20358
In this paper, by studying three $q$-Catalan identities given by Andrews, we arrive at a certain number of congruences. These congruences are all modulo $Phi_n(q)$, the $n$-th cyclotomic polynomial or the related functions and modulo $q$-integers.
Congruences,$q$-Catalan identities,Catalan numbers,$q$-integer,Cyclotomic polynomial
http://toc.ui.ac.ir/article_20358.html
http://toc.ui.ac.ir/article_20358_742244d2cadb0585b9b1cc7a3cde94c5.pdf