ORIGINAL_ARTICLE
Cacti with extremal PI Index
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
2016-12-01T11:23:20
2017-12-11T11:23:20
1
8
10.22108/toc.2016.14786
Distance
Extremal bounds
PI index
Cacti
Chunxiang
Wang
wcxiang@mail.ccnu.edu.cn
true
1
Central China Normal University
Central China Normal University
Central China Normal University
AUTHOR
Shaohui
Wang
shaohuiwang@yahoo.com
true
2
University of Mississippi
University of Mississippi
University of Mississippi
LEAD_AUTHOR
Bing
Wei
bwei@olemiss.edu
true
3
University of Mississippi
University of Mississippi
University of Mississippi
AUTHOR
[1] T. Al-Fozan, P. Manuel, I. Rajasingh and R. S. Rajan, Computing Szeged index of certain nanosheets using partition technique, MATCH Commun. Math. Comput. Chem., 72 (2014) 339–353.
1
[2] A. R. Ashrafi and A. Loghman, PI index of zig-zag polyhex nanotubes, MATCH Commun. Math. Comput. Chem., 55 (2006) 447–452.
2
[3] A. R. Ashrafi and A. Loghman, Padmakar-Ivan index of TUC4C8(S) nanotubes, J. Comput. Theor. Nanosci., 3 (2006) 378–381.
3
[4] A. R. Ashrafi and A. Loghman, PI index of armchair polyhex nanotubes, Ars Combin., 80 (2006) 193–199.
4
[5] A. R. Ashrafi, B. Manoochehrian and H. Yousefi-Azari, On the PI polynomial of a graph, Util. Math., 71 (2006) 97–108.
5
[6] A. R. Ashrafi and F. Rezaei, PI index of polyhex nanotori, MATCH Commun. Math. Comput. Chem., 57 (2007) 243–250.
6
[7] S. Chen, Cacti with the smallest, second smallest and third smallest Gutman index, J. Comb. Optim., 31 (2016) 327–332.
7
[8] A. A. Dobrynin, R. Entringer and I. Gutman, Wiener index of trees: theory and applications, Acta Appl. Math., 66 (2001) 211–249.
8
[9] A. A. Dobrynin, I. Gutman, S. Klavzzar and P. zZigert, Wiener index of hexagonal systems, Acta Appl. Math., 72 (2002) 247–294.
9
[10] K. C. Das and I. Gutman, Bound for vertex PI index in terms of simple graph parameters, Filomat, 27 (2013) 1583–1587.
10
[11] L. Feng and G. Yu, On the hyper-Wiener index of cacti, Util. Math., 93 (2014) 57–64.
11
[12] I. Gutman, S. Klavzar and B. Mohar, Fiftieth Anniversary of the Wiener Index, Discrete Appl. Math., 80 (1997) 1–113.
12
[13] M. Hoji, Z. Luo and E. Vumar, Wiener and vertex PI indices of Kronecker products of graphs, Discrete Appl. Math., 158 (2010) 1848–1855.
13
[14] A. Ilic and N. Milosavljevic, The weighted vertex PI index, Mathematical and Computer Modelling., 57 (2013) 623–631.
14
[15] S. Klavzar and I. Gutman, The Szeged and the Wiener Index of Graphs, Appl. Math. Lett., 9 (1996) 45–49.
15
[16] P. V. Khadikar, On a Novel Structural Descriptor PI, Nat. Acad. Sci. Lett., 23 (2000) 113–118.
16
[17] P. V. Khadikar, P. P. Kale, N. V. Deshpande, S. Karmarkar and V. K. Agrawal, Novel PI indices of hexagonal chains, J. Math. Chem., 29 (2001) 143–150.
17
[18] P. V. Khadikar, S. Karmarkar and R. G. Varma, The estimation of PI index of polyacenes, Acta Chim. Slov., 49 (2002) 755–771.
18
[19] M. H. Khalifeh, H. Yousefi-Azari and A. R. Ashrafi, Vertex and edge PI indices of Cartesian product graphs, Discrete Appl. Math., 156 (2008) 1780–1789.
19
[20] S. Li, H. Yang and Q. Zhao, Sharp bounds on Zagreb indices of cacti with $k$ pendant vertices, Filomat, 26 (2012) 1189–1200.
20
[21] K. Pattabiraman and P. Paulraja, Wiener and vertex PI indices of the strong product of graphs, Discuss. Math. Graph Theory, 32 (2012) 749–769.
21
[22] D. Wang and S. Tan, The maximum hyper-Wiener index of cacti, J. Appl. Math. Comput., 47 (2015) 91–102.
22
[23] H. Wang and L. Kang, On the Harary index of cacti, Util. Math., 96 (2015) 149–163.
23
[24] H. Wiener, Structural Determination of Paraffin Boiling Points, J. Am. Chem. Soc., 69 (1947) 17–20.
24
[25] S. Wang and B. Wei, Multiplicative Zagreb indices of cacti, Discrete Math. Algorithm. Appl.,
25
DOI:10.1142/S1793830916500403.
26
[26] S. Wang and B. Wei, Multiplicative Zagreb indices of $k$-trees, Discrete Appl. Math., 180 (2015) 168–175.
27
ORIGINAL_ARTICLE
Some results on the comaximal ideal graph of a commutative ring
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
2016-12-01T11:23:20
2017-12-11T11:23:20
9
20
10.22108/toc.2016.15047
Comaximal ideal graph
Genus of graph
Domination Number
Independence number
Hamid Reza
Dorbidi
hr_dorbidi@yahoo.com
true
1
University of Jiroft,Jiroft, Kerman, Iran
University of Jiroft,Jiroft, Kerman, Iran
University of Jiroft,Jiroft, Kerman, Iran
LEAD_AUTHOR
Raoufeh
Manaviyat
r.manaviyat@gmail.com
true
2
Payame Noor University, Tehran, Iran
Payame Noor University, Tehran, Iran
Payame Noor University, Tehran, Iran
AUTHOR
[1] S. Akbari, M. Habibi, A. Majidinya and R. Manaviyat, A note on comaximal graph of non-commutative rings, Algebr. Represent. Theory, 16 no. 2 (2013) 303–307.
1
[2] S. Akbari, M. Habibi, A. Majidinya and R. Manaviyat, On the idempotent graph of a ring, J. Algebra Appl., 12 no. 6 (2013) pp. 14.
2
[3] S. Akbari, B. Miraftab and R. Nikandish, A Note on Co-Maximal Ideal Graph of Commutative Rings, Ars Combin., To appear.
3
[4] S. Akbari and R. Nikandish, Some results on the intersection graphs of ideals of matrix algebras, Linear and Multilinear Algebra, 62 no. 2 (2014) 195–206.
4
[5] D. Archdeacon, Topological graph theory: a survey, Congr. Numer., 115 (1996) 5–54.
5
[6] M. I. Jinnah and S. C. Mathew, When is the comaximal graph split?, Comm. Algebra, 40 no. 7 (2012) 2400–2404.
6
[7] H .R. Maimani, M. Salimi, A. Sattari and S. Yassemi, Comaximal graph of commutative rings, J. Algebra, 319 no. 4 (2008) 1801–1808.
7
[8] P. K. Sharma and S. M. Bhatwadekar, A note on graphical representation of rings, J. Algebra, 176 no. 1 (1995) 124–127.
8
[9] H. J. Wang, Graphs associated to co-maximal ideals of commutative rings, J. Algebra, 320 no. 7 (2008) 2917–2933.
9
[10] A. T. White, Graphs, Groups and Surfaces, North-Holland Mathematics Studies, North-Holland, Amsterdam, 1973.
10
[11] T. Wu and M. Ye, Co-maximal ideal graphs of commutative rings, J. Algebra Appl., 11 no. 6 (2012) pp. 14.
11
[12] T. Wu, M. Ye, Q. Liu and J. Guo, Graph properties of co-maximal ideal graphs of commutative rings, J. Algebra Appl., 14 no. 3 (2015) pp. 13.
12
[13] M. Ye, T. S. Wu, Q. Liu and H. Yu, Implements of Graph Blow-Up in Co-Maximal Ideal Graphs, Comm. Algebra, 42 no. 6 (2014) 2476–2483.
13
ORIGINAL_ARTICLE
On the new extension of distance-balanced graphs
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
2016-12-01T11:23:20
2017-12-11T11:23:20
21
34
10.22108/toc.2016.15048
$n$-distance-balanced property
lexicographic product
total distance
Morteza
Faghani
m_faghani@pnu.ac.ir
true
1
Chief of PNU Saveh branch
Chief of PNU Saveh branch
Chief of PNU Saveh branch
LEAD_AUTHOR
Ehsan
Pourhadi
epourhadi@iust.ac.ir
true
2
Comprehensive Imam Hossein University
Comprehensive Imam Hossein University
Comprehensive Imam Hossein University
AUTHOR
Hassan
Kharazi
hkharazi@ihu.ac.ir
true
3
Comprehensive Imam Hossein University
Comprehensive Imam Hossein University
Comprehensive Imam Hossein University
AUTHOR
[1] K. Balakrishnan, M. Changat, I. Peterin, S. Spacapan, P. Sparl and A. R. Subhamathi, Strongly distance-balanced graphs and graph products, European J. Combin., 30 no. 5 (2009) 1048–1053.
1
[2] F. Buckley and F. Harary, Distance in Graphs, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1990.
2
[3] S. Cabello and P. Luksic, The complexity of obtaining a distance-balanced graph, Electron. J. Combin., 18 no. 1 (2011) 10 pp. 34
3
[4] L. Chen, X. Li, M. Liu and I. Gutman, On a relation between Szeged and Wiener indices of bipartite graphs,Trans. Combin., 1 no. 4 (2012) 43–49.
4
[5] A. A. Dobrynin, R. Entringer and I. Gutman, Wiener index of trees: theory and applications, Acta Appl. Math., 66 no. 3 (2001) 211–249.
5
[6] K. Fukuda and K. Handa, Antipodal graphs and oriented matroids, Discrete Math., 111 no. 1-3 (1993) 245–256.
6
[7] J. A. Gallian, Dynamic Survey DS6: Graph Labeling, Electronic J. Combin., DS6, (2007) 1–58.
7
[8] I. Gutman and A. A. Dobrynin, The Szeged index-a success story, Graph Theory Notes N. Y., 34 (1998) 37–44.
8
[9] K. Handa, Bipartite graphs with balanced (a,b)-partitions, Ars Combin., 51 (1999) 113–119.
9
[10] A. Ilic, S. Klavzar and M. Milanovic, On distance-balanced graphs, European J. Combin., 31 no. 3 (2010) 733–737.
10
[11] J. Jerebic, S. Klavzar and D. F. Rall, Distance-balanced graphs, Ann. Combin., 12 no. 1 (2008) 71–79.
11
[12] M. H. Khalifeh, H. Yousefi-Azari, A. R. Ashrafi and S. G. Wagner, Some new results on distance-based graph invariants, European J. Combin., 30 (2009) 1149–1163.
12
[13] K. Kutnar, A. Malnic, D. Marusic and S. Miklavic, Distance-balanced graphs: Symmetry conditions, Discrete Math., 306 (2006) 1881–1894.
13
[14] K. Kutnar, A. Malnic, D. Marusic and S. Miklavic, The strongly distance-balanced property of the generalized Petersen graphs, Ars Math. Contemp., 2 no. 1 (2009) 41–47.
14
[15] K. Kutnar and S. Miklavic, Nicely distance-balanced graphs, European J. Combin., 39 (2014) 57–67.
15
[16] S. Miklavic and P. Sparl, On the connectivity of bipartite distance-balanced graphs, European J. Combin., 33 no. 2 (2012) 237–247.
16
[17] M. Tavakoli, H. Yousefi-Azari and A. R. Ashrafi, Note on edge distance-balanced graphs, Trans. Combin., 1 no. 1 (2012) 1–6.
17
[18] R. Yang, X. Hou, N. Li and W. Zhong, A note on the distance-balanced property of generalized Petersen graphs, Electron. J. Combin., 16 no. 1 (2009) 3 pp.
18
ORIGINAL_ARTICLE
Extremal tetracyclic graphs with respect to the first and second Zagreb indices
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.
http://toc.ui.ac.ir/article_12878_b7525583ad7d958b2f5cb6c2d9eabfdb.pdf
2016-12-01T11:23:20
2017-12-11T11:23:20
35
55
10.22108/toc.2016.12878
First Zagreb index
second Zagreb index
tetracyclic graph
Nader
Habibi
nader.habibi@ymail.com
true
1
university of Ayatollah Al-ozma
university of Ayatollah Al-ozma
university of Ayatollah Al-ozma
LEAD_AUTHOR
Tayebeh
Dehghan Zadeh
ta.dehghanzadeh@gmail.com
true
2
University of Kashan
University of Kashan
University of Kashan
AUTHOR
Ali Reza
Ashrafi
ashrafi@kashan.ac.ir
true
3
University of Kashan
University of Kashan
University of Kashan
AUTHOR
ORIGINAL_ARTICLE
Congruences from $q$-Catalan Identities
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
2016-12-01T11:23:20
2017-12-11T11:23:20
57
67
10.22108/toc.2016.20358
Congruences
$q$-Catalan identities
Catalan numbers
$q$-integer
Cyclotomic polynomial
Qing
Zou
zou-qing@uiowa.edu
true
1
Department of Mathematics, The University of Iowa
Department of Mathematics, The University of Iowa
Department of Mathematics, The University of Iowa
LEAD_AUTHOR