2016
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Cacti with extremal PI Index
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2
The vertex PI index $PI(G) = sum_{xy in E(G)} [n_{xy}(x) + n_{xy}(y)]$ is a distancebased 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.
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1
8


Chunxiang
Wang
Central China Normal University
Central China Normal University
China
wcxiang@mail.ccnu.edu.cn


Shaohui
Wang
University of Mississippi
University of Mississippi
United States of America
shaohuiwang@yahoo.com


Bing
Wei
University of Mississippi
University of Mississippi
United States of America
bwei@olemiss.edu
Distance
Extremal bounds
PI index
Cacti
[[1] T. AlFozan, 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.##[2] A. R. Ashrafi and A. Loghman, PI index of zigzag polyhex nanotubes, MATCH Commun. Math. Comput. Chem., 55 (2006) 447–452.##[3] A. R. Ashrafi and A. Loghman, PadmakarIvan index of TUC4C8(S) nanotubes, J. Comput. Theor. Nanosci., 3 (2006) 378–381.##[4] A. R. Ashrafi and A. Loghman, PI index of armchair polyhex nanotubes, Ars Combin., 80 (2006) 193–199.##[5] A. R. Ashrafi, B. Manoochehrian and H. YousefiAzari, On the PI polynomial of a graph, Util. Math., 71 (2006) 97–108.##[6] A. R. Ashrafi and F. Rezaei, PI index of polyhex nanotori, MATCH Commun. Math. Comput. Chem., 57 (2007) 243–250.##[7] S. Chen, Cacti with the smallest, second smallest and third smallest Gutman index, J. Comb. Optim., 31 (2016) 327–332.##[8] A. A. Dobrynin, R. Entringer and I. Gutman, Wiener index of trees: theory and applications, Acta Appl. Math., 66 (2001) 211–249.##[9] A. A. Dobrynin, I. Gutman, S. Klavzzar and P. zZigert, Wiener index of hexagonal systems, Acta Appl. Math., 72 (2002) 247–294.##[10] K. C. Das and I. Gutman, Bound for vertex PI index in terms of simple graph parameters, Filomat, 27 (2013) 1583–1587.##[11] L. Feng and G. Yu, On the hyperWiener index of cacti, Util. Math., 93 (2014) 57–64.##[12] I. Gutman, S. Klavzar and B. Mohar, Fiftieth Anniversary of the Wiener Index, Discrete Appl. Math., 80 (1997) 1–113.##[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.##[14] A. Ilic and N. Milosavljevic, The weighted vertex PI index, Mathematical and Computer Modelling., 57 (2013) 623–631.##[15] S. Klavzar and I. Gutman, The Szeged and the Wiener Index of Graphs, Appl. Math. Lett., 9 (1996) 45–49.##[16] P. V. Khadikar, On a Novel Structural Descriptor PI, Nat. Acad. Sci. Lett., 23 (2000) 113–118.##[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.##[18] P. V. Khadikar, S. Karmarkar and R. G. Varma, The estimation of PI index of polyacenes, Acta Chim. Slov., 49 (2002) 755–771.##[19] M. H. Khalifeh, H. YousefiAzari and A. R. Ashrafi, Vertex and edge PI indices of Cartesian product graphs, Discrete Appl. Math., 156 (2008) 1780–1789.##[20] S. Li, H. Yang and Q. Zhao, Sharp bounds on Zagreb indices of cacti with $k$ pendant vertices, Filomat, 26 (2012) 1189–1200.##[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.##[22] D. Wang and S. Tan, The maximum hyperWiener index of cacti, J. Appl. Math. Comput., 47 (2015) 91–102.##[23] H. Wang and L. Kang, On the Harary index of cacti, Util. Math., 96 (2015) 149–163.##[24] H. Wiener, Structural Determination of Paraffin Boiling Points, J. Am. Chem. Soc., 69 (1947) 17–20.##[25] S. Wang and B. Wei, Multiplicative Zagreb indices of cacti, Discrete Math. Algorithm. Appl.,##DOI:10.1142/S1793830916500403.##[26] S. Wang and B. Wei, Multiplicative Zagreb indices of $k$trees, Discrete Appl. Math., 180 (2015) 168–175.##]
Some results on the comaximal ideal graph of a commutative ring
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2
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 $mathcal{C}(mathbb{Z}_n)$.
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9
20


Hamid Reza
Dorbidi
University of Jiroft,Jiroft, Kerman, Iran
University of Jiroft,Jiroft, Kerman, Iran
Iran
hr_dorbidi@yahoo.com


Raoufeh
Manaviyat
Payame Noor University, Tehran, Iran
Payame Noor University, Tehran, Iran
Iran
r.manaviyat@gmail.com
Comaximal ideal graph
Genus of graph
Domination Number
Independence number
[[1] S. Akbari, M. Habibi, A. Majidinya and R. Manaviyat, A note on comaximal graph of noncommutative rings, Algebr. Represent. Theory, 16 no. 2 (2013) 303–307.##[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.##[3] S. Akbari, B. Miraftab and R. Nikandish, A Note on CoMaximal Ideal Graph of Commutative Rings, Ars Combin., To appear.##[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.##[5] D. Archdeacon, Topological graph theory: a survey, Congr. Numer., 115 (1996) 5–54.##[6] M. I. Jinnah and S. C. Mathew, When is the comaximal graph split?, Comm. Algebra, 40 no. 7 (2012) 2400–2404.##[7] H .R. Maimani, M. Salimi, A. Sattari and S. Yassemi, Comaximal graph of commutative rings, J. Algebra, 319 no. 4 (2008) 1801–1808.##[8] P. K. Sharma and S. M. Bhatwadekar, A note on graphical representation of rings, J. Algebra, 176 no. 1 (1995) 124–127.##[9] H. J. Wang, Graphs associated to comaximal ideals of commutative rings, J. Algebra, 320 no. 7 (2008) 2917–2933.##[10] A. T. White, Graphs, Groups and Surfaces, NorthHolland Mathematics Studies, NorthHolland, Amsterdam, 1973.##[11] T. Wu and M. Ye, Comaximal ideal graphs of commutative rings, J. Algebra Appl., 11 no. 6 (2012) pp. 14.##[12] T. Wu, M. Ye, Q. Liu and J. Guo, Graph properties of comaximal ideal graphs of commutative rings, J. Algebra Appl., 14 no. 3 (2015) pp. 13. ##[13] M. Ye, T. S. Wu, Q. Liu and H. Yu, Implements of Graph BlowUp in CoMaximal Ideal Graphs, Comm. Algebra, 42 no. 6 (2014) 2476–2483.##]
On the new extension of distancebalanced graphs
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In this paper, we initially introduce the concept of $n$distancebalanced property which is considered as the generalized concept of distancebalanced property. In our consideration, we also define the new concept locally regularity in order to find a connection between $n$distancebalanced graphs and their lexicographic product. Furthermore, we include a characteristic method which is practicable and can be used to classify all graphs with $i$distancebalanced properties for $ i=2,3 $ which is also relevant to the concept of total distance. Moreover, we conclude a connection between distancebalanced and 2distancebalanced graphs.
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Morteza
Faghani
Chief of PNU Saveh branch
Chief of PNU Saveh branch
Iran
m_faghani@pnu.ac.ir


Ehsan
Pourhadi
Comprehensive Imam Hossein University
Comprehensive Imam Hossein University
Iran
epourhadi@iust.ac.ir


Hassan
Kharazi
Comprehensive Imam Hossein University
Comprehensive Imam Hossein University
Iran
hkharazi@ihu.ac.ir
$n$distancebalanced property
lexicographic product
total distance
[[1] K. Balakrishnan, M. Changat, I. Peterin, S. Spacapan, P. Sparl and A. R. Subhamathi, Strongly distancebalanced graphs and graph products, European J. Combin., 30 no. 5 (2009) 1048–1053.##[2] F. Buckley and F. Harary, Distance in Graphs, AddisonWesley Publishing Company, Advanced Book Program, Redwood City, CA, 1990.##[3] S. Cabello and P. Luksic, The complexity of obtaining a distancebalanced graph, Electron. J. Combin., 18 no. 1 (2011) 10 pp. 34##[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.##[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.##[6] K. Fukuda and K. Handa, Antipodal graphs and oriented matroids, Discrete Math., 111 no. 13 (1993) 245–256.##[7] J. A. Gallian, Dynamic Survey DS6: Graph Labeling, Electronic J. Combin., DS6, (2007) 1–58.##[8] I. Gutman and A. A. Dobrynin, The Szeged indexa success story, Graph Theory Notes N. Y., 34 (1998) 37–44.##[9] K. Handa, Bipartite graphs with balanced (a,b)partitions, Ars Combin., 51 (1999) 113–119.##[10] A. Ilic, S. Klavzar and M. Milanovic, On distancebalanced graphs, European J. Combin., 31 no. 3 (2010) 733–737.##[11] J. Jerebic, S. Klavzar and D. F. Rall, Distancebalanced graphs, Ann. Combin., 12 no. 1 (2008) 71–79.##[12] M. H. Khalifeh, H. YousefiAzari, A. R. Ashrafi and S. G. Wagner, Some new results on distancebased graph invariants, European J. Combin., 30 (2009) 1149–1163.##[13] K. Kutnar, A. Malnic, D. Marusic and S. Miklavic, Distancebalanced graphs: Symmetry conditions, Discrete Math., 306 (2006) 1881–1894.##[14] K. Kutnar, A. Malnic, D. Marusic and S. Miklavic, The strongly distancebalanced property of the generalized Petersen graphs, Ars Math. Contemp., 2 no. 1 (2009) 41–47.##[15] K. Kutnar and S. Miklavic, Nicely distancebalanced graphs, European J. Combin., 39 (2014) 57–67.##[16] S. Miklavic and P. Sparl, On the connectivity of bipartite distancebalanced graphs, European J. Combin., 33 no. 2 (2012) 237–247.##[17] M. Tavakoli, H. YousefiAzari and A. R. Ashrafi, Note on edge distancebalanced graphs, Trans. Combin., 1 no. 1 (2012) 1–6.##[18] R. Yang, X. Hou, N. Li and W. Zhong, A note on the distancebalanced property of generalized Petersen graphs, Electron. J. Combin., 16 no. 1 (2009) 3 pp.##]
Extremal tetracyclic graphs with respect to the first and second Zagreb indices
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2
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.
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55


Nader
Habibi
university of Ayatollah Alozma
university of Ayatollah Alozma
Iran
nader.habibi@ymail.com


Tayebeh
Dehghan Zadeh
University of Kashan
University of Kashan
Iran
ta.dehghanzadeh@gmail.com


Ali Reza
Ashrafi
University of Kashan
University of Kashan
Iran
ashrafi@kashan.ac.ir
First Zagreb index
second Zagreb index
tetracyclic graph
Congruences from $q$Catalan Identities
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2
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.
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57
67


Qing
Zou
Department of Mathematics, The University of Iowa
Department of Mathematics, The University
China
zouqing@uiowa.edu
Congruences
$q$Catalan identities
Catalan numbers
$q$integer
Cyclotomic polynomial