eng
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2016-12-01
5
4
1
8
10.22108/toc.2016.14786
14786
Cacti with extremal PI Index
Chunxiang Wang
wcxiang@mail.ccnu.edu.cn
1
Shaohui Wang
shaohuiwang@yahoo.com
2
Bing Wei
bwei@olemiss.edu
3
Central China Normal University
University of Mississippi
University of Mississippi
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.
http://toc.ui.ac.ir/article_14786_f95e820e8bf0d1325600f95c8a3d7a24.pdf
Distance
Extremal bounds
PI index
Cacti
eng
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2016-12-01
5
4
9
20
10.22108/toc.2016.15047
15047
Some results on the comaximal ideal graph of a commutative ring
Hamid Reza Dorbidi
hr_dorbidi@yahoo.com
1
Raoufeh Manaviyat
r.manaviyat@gmail.com
2
University of Jiroft,Jiroft, Kerman, Iran
Payame Noor University, Tehran, Iran
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 $mathcal{C}(mathbb{Z}_n)$.
http://toc.ui.ac.ir/article_15047_e2760f540dc55e62152260c257848270.pdf
Comaximal ideal graph
Genus of graph
Domination Number
Independence number
eng
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2016-12-01
5
4
21
34
10.22108/toc.2016.15048
15048
On the new extension of distance-balanced graphs
Morteza Faghani
m_faghani@pnu.ac.ir
1
Ehsan Pourhadi
epourhadi@iust.ac.ir
2
Hassan Kharazi
hkharazi@ihu.ac.ir
3
Chief of PNU Saveh branch
Comprehensive Imam Hossein University
Comprehensive Imam Hossein University
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.
http://toc.ui.ac.ir/article_15048_3968109258ac5aaddf5a16c03fc677d5.pdf
$n$-distance-balanced property
lexicographic product
total distance
eng
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2016-12-01
5
4
35
55
10.22108/toc.2016.12878
12878
Extremal tetracyclic graphs with respect to the first and second Zagreb indices
Nader Habibi
nader.habibi@ymail.com
1
Tayebeh Dehghan Zadeh
ta.dehghanzadeh@gmail.com
2
Ali Reza Ashrafi
ashrafi@kashan.ac.ir
3
university of Ayatollah Al-ozma
University of Kashan
University of Kashan
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.
http://toc.ui.ac.ir/article_12878_b7525583ad7d958b2f5cb6c2d9eabfdb.pdf
First Zagreb index
second Zagreb index
tetracyclic graph
eng
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2016-12-01
5
4
57
67
10.22108/toc.2016.20358
20358
Congruences from $q$-Catalan Identities
Qing Zou
zou-qing@uiowa.edu
1
Department of Mathematics, The University of Iowa
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.
http://toc.ui.ac.ir/article_20358_742244d2cadb0585b9b1cc7a3cde94c5.pdf
Congruences
$q$-Catalan identities
Catalan numbers
$q$-integer
Cyclotomic polynomial