Cacti with extremal PI Index
Chunxiang
Wang
Central China Normal University
author
Shaohui
Wang
University of Mississippi
author
Bing
Wei
University of Mississippi
author
text
article
2016
eng
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.
Transactions on Combinatorics
University of Isfahan
2251-8657
5
v.
4
no.
2016
1
8
http://toc.ui.ac.ir/article_14786_f95e820e8bf0d1325600f95c8a3d7a24.pdf
dx.doi.org/10.22108/toc.2016.14786
Some results on the comaximal ideal graph of a commutative ring
Hamid Reza
Dorbidi
University of Jiroft,Jiroft, Kerman, Iran
author
Raoufeh
Manaviyat
Payame Noor University, Tehran, Iran
author
text
article
2016
eng
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 $\mathcal{C}(\mathbb{Z}_n)$.
Transactions on Combinatorics
University of Isfahan
2251-8657
5
v.
4
no.
2016
9
20
http://toc.ui.ac.ir/article_15047_e2760f540dc55e62152260c257848270.pdf
dx.doi.org/10.22108/toc.2016.15047
On the new extension of distance-balanced graphs
Morteza
Faghani
Chief of PNU Saveh branch
author
Ehsan
Pourhadi
Comprehensive Imam Hossein University
author
Hassan
Kharazi
Comprehensive Imam Hossein University
author
text
article
2016
eng
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.
Transactions on Combinatorics
University of Isfahan
2251-8657
5
v.
4
no.
2016
21
34
http://toc.ui.ac.ir/article_15048_3968109258ac5aaddf5a16c03fc677d5.pdf
dx.doi.org/10.22108/toc.2016.15048
Extremal tetracyclic graphs with respect to the first and second Zagreb indices
Nader
Habibi
university of Ayatollah Al-ozma
author
Tayebeh
Dehghan Zadeh
University of Kashan
author
Ali Reza
Ashrafi
University of Kashan
author
text
article
2016
eng
The first Zagreb index, $M_1(G)$, and second Zagreb index, $M_2(G)$, of the graph $G$ is defined as $M_{1}(G)=\sum_{v\in V(G)}d^{2}(v)$ and $M_{2}(G)=\sum_{e=uv\in E(G)}d(u)d(v),$ 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.
Transactions on Combinatorics
University of Isfahan
2251-8657
5
v.
4
no.
2016
35
55
http://toc.ui.ac.ir/article_12878_b7525583ad7d958b2f5cb6c2d9eabfdb.pdf
dx.doi.org/10.22108/toc.2016.12878
Congruences from $q$-Catalan Identities
Qing
Zou
Department of Mathematics, The University of Iowa
author
text
article
2016
eng
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.
Transactions on Combinatorics
University of Isfahan
2251-8657
5
v.
4
no.
2016
57
67
http://toc.ui.ac.ir/article_20358_742244d2cadb0585b9b1cc7a3cde94c5.pdf
dx.doi.org/10.22108/toc.2016.20358