2018-09-21T22:54:09Z
http://toc.ui.ac.ir/?_action=export&rf=summon&issue=2938
Transactions on Combinatorics
Trans. Comb.
2251-8657
2251-8657
2016
5
4
Cacti with extremal PI Index
Chunxiang
Wang
Shaohui
Wang
Bing
Wei
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
2016
12
01
1
8
http://toc.ui.ac.ir/article_14786_f95e820e8bf0d1325600f95c8a3d7a24.pdf
Transactions on Combinatorics
Trans. Comb.
2251-8657
2251-8657
2016
5
4
Some results on the comaximal ideal graph of a commutative ring
Hamid Reza
Dorbidi
Raoufeh
Manaviyat
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
2016
12
01
9
20
http://toc.ui.ac.ir/article_15047_e2760f540dc55e62152260c257848270.pdf
Transactions on Combinatorics
Trans. Comb.
2251-8657
2251-8657
2016
5
4
On the new extension of distance-balanced graphs
Morteza
Faghani
Ehsan
Pourhadi
Hassan
Kharazi
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
2016
12
01
21
34
http://toc.ui.ac.ir/article_15048_3968109258ac5aaddf5a16c03fc677d5.pdf
Transactions on Combinatorics
Trans. Comb.
2251-8657
2251-8657
2016
5
4
Extremal tetracyclic graphs with respect to the first and second Zagreb indices
Nader
Habibi
Tayebeh
Dehghan Zadeh
Ali Reza
Ashrafi
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
2016
12
01
35
55
http://toc.ui.ac.ir/article_12878_b7525583ad7d958b2f5cb6c2d9eabfdb.pdf
Transactions on Combinatorics
Trans. Comb.
2251-8657
2251-8657
2016
5
4
Congruences from $q$-Catalan Identities
Qing
Zou
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
2016
12
01
57
67
http://toc.ui.ac.ir/article_20358_742244d2cadb0585b9b1cc7a3cde94c5.pdf