2018
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4
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59
The Extremal Graphs for (Sum) Balaban Index of Spiro and Polyphenyl Hexagonal Chains
2
2
As highly discriminant distancebased topological indices, the Balaban index and the sumBalaban index of a graph $G$ are defined as $J(G)=frac{m}{mu+1}sumlimits_{uvin E} frac{1}{sqrt{D_{G}(u)D_{G}(v)}}$ and $SJ(G)=frac{m}{mu+1}sumlimits_{uvin E} frac{1}{sqrt{D_{G}(u)+D_{G}(v)}}$, respectively, where $D_{G}(u)=sumlimits_{vin V}d(u,v)$ is the distance sum of vertex $u$ in $G$, $m$ is the number of edges and $mu$ is the cyclomatic number of $G$. They are useful distancebased descriptor in chemometrics. In this paper, we focus on the extremal graphs of spiro and polyphenyl hexagonal chains with respect to the Balaban index and the sumBalaban index.
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241
254


Y.
Zuo
College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China
College of Mathematics and Statistics, Hunan
P. R. China
yzuo@163.com


Y.
Tang
College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China
College of Mathematics and Statistics, Hunan
P. R. China
tang015@163.com


H. Y.
Deng
College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China
College of Mathematics and Statistics, Hunan
P. R. China
hydeng@hunnu.edu.cn
Balaban index
sumBalaban index
spiro hexagonal chain, polyphenyl hexagonal chain
[1. V. Andová, M. Knor, R. Škrekovski, Distance based indices in nanotubical graphs:part 2, J. Math. Chem. DOI: 10.1007/s1091001809332.##2. Y. Bai, B. Zhao, P. Zhao, Extremal MerrifildSimmons index and Hosoya index of polyphenyl chains, MATCH Commun. Math. Comput. Chem. 62 (2009) 649–656.##3. A. T. Balaban, Highly discriminating distancebased topological index, Chem. Phys. Lett. 89 (1982) 399–404.##4. A. T. Balaban, Topological index based on topological distance in molecular graphs, Pure Appl. Chem. 55 (1983) 199–206.##5. A. T. Balaban, P. V. Khadikar, S. Aziz, Comparison of topological index based on iterated ’sum’ versus ’product’ operations, Iranian J. Math. Chem. 1 (2010) 43–67.##6. A. T. Balaban, L. B. Kier, N. Joshi, Structureproperty analysis of octane numbers for hydrocarbons (alkanes, cycloalkanes, alkenes), MATCH Commun. Math. Comput. Chem. 28 (1992) 13–27.##7. D. Bonchev, E. J. Markel, A. H. Dekmezian, Long chain branch polymer chain dimensions: application of topology to the ZimmStockmayer model, Polymer 43 (2002) 203–222.##8. M. Bureš, V. Pekárek, T. Ocelka, Thermochemical properties and relative stability of polychlorinated biphenyls, Environ. Tox. Pharm. 25 (2008) 2610–2617.##9. X. Chen, B. Zhao, P. Zhao, Sixmembered ring spiro chains with extremal MerrifildSimmons index and Hosoya index, MATCH Commun. Math. Comput. Chem. 62 (2009) 657–665.##10. H. Deng, On the Balaban index of trees, MATCH Commun. Math. Comput. Chem. 66 (2011) 253–260.##11. H. Deng, On the SumBalaban index, MATCH Commun. Math. Comput. Chem. 66 (2011) 273–284.##12. H. Deng, Wiener indices of spiro and polyphenyl hexagonal chains, Math. Comput. Modelling 55 (2012) 634–644.##13. H. Deng, Z. Tang, Kirchhoff indices of spiro and polyphenyl hexagonal chains, Util. Math. 95 (2014) 113–128.##14. H. Dong, X. Guo, Character of graphs with extremal Balaban index, MATCH Commun. Math. ComputChem. 63 (2010) 799–812.##15. H. Dong, X. Guo, Character of trees with extremal Balaban index, MATCH Commun. Math. Comput. Chem. 66 (2011) 261–272.##16. T. Došlić, M. Litz, Matchings and independent sets in polyphenylene chains, MATCH Commun. Math. Comput. Chem. 67 (2012) 313–330.##17. T. Došlić, F. Måløy, Chain hexagonal cacti: Matchings and independent sets, Discrete Math. 310 (2010) 1676–1690.##18. W. Fang, Y. Gao, Y. Shao, W. Gao, G. Jing, Z. Li, Maximum Balaban index and SumBalaban index of bicyclic graphs, MATCH Commun. Math. Comput. Chem. 75 (2016) 129–156.##19. W. Fang, H. Yu, Y. Gao, X. Li, G. Jing, Z. Li, Maximum Balaban index and SumBalaban index of tricyclic graphs, MATCH Commun. Math. Comput. Chem. 79 (2018) 717–742.##20. D. R. Flower, On the properties of bit stringbased measures of chemical similarity, J. Chem. Inf. Comput. Sci. 38 (1998) 379–386.##21. A. Graja, LowDimensional Organic Conductors, World Scientific, Singapore, 1992.##22. G. Grassy, B. Calas, A. Yasri, R. Lahana, J. Woo, S. Iyer, M. Kaczorek, F. Floch, R. Buelow, Computerassisted rational design of immunosuppressive compounds, Nat. Biotechnol. 16 (1998) 748–752.##23. G. Huang, M. Kuang, H. Deng, The extremal graph with respect to the matching energy for a random polyphenyl chain, MATCH Commun. Math. Comput. Chem. 73 (2015) 121–131.##24. G. Huang, M. Kuang, H. Deng, The expected values of Hosoya index and MerrifieldSimmons index in a random polyphenylence chain, J. Combin. Opt. 32 (2) (2016) 550–562.##25. M. Knor, R. Škrekovski, A. Tepeh, Balaban index of cubic graphs, MATCH Commun. Math. Comput. Chem. 73 (2015) 519–528.##26. Q. R. Li, Q. Yang, H. Yin, S. Yang, Analysis of byproducts from improved Ullmann reaction using TOFMS and GCTOFMS, J. Univ. Sci. Technol. China 34 (2004) 335–341.##27. S. Li, B. Zhou, On the Balaban index of trees, Ars Combin. 101 (2011) 503–512.##28. G. Luthe, J. A. Jacobus, L. W. Robertson, Receptor interactions by polybrominated diphenyl ethers versus polychlorinated biphenyls: A theoretical structureactivity assessment, Environ. Tox. Pharm. 25 (2008) 202–210.##29. L. Sun, Bounds on the Balaban index of trees, MATCH Commun. Math. Comput. Chem. 63 (2010) 813–818.##30. S. Tepavčević, A. T. Wroble, M. Bissen, D. J. Wallace, Y. Choi, L. Hanley, Photoemission studies of polythiophene and polyphenyl films produced via surface polymerization by ionassisted deposition, J. Phys. Chem. B 109 (2005) 7134–7140.##31. R. Xing, B. Zhou, A. Graovac, On SumBalaban index, Ars Combin. 104 (2012) 211–223.##32. W. Yang, F. Zhang, Wiener index in random polyphenyl chains, MATCH Commun. Math. Comput. Chem. 68 (2012) 371–376.##33. L. You, X. Dong, The maximum Balaban index (SumBalaban index) of unicyclic graphs, J. Math. Res. Appl. 34 (2014) 292–402.##34. L. You, H. Han, The maximum Balaban index (SumBalaban index) of trees with given diameter, Ars Combin. 112 (2013) 115–128.##35. P. Zhao, B. Zhao, X. Chen, Y. Bai, Two classes of chains with maximal and minmal total electron energy, MATCH Commun. Math. Comput. Chem. 62 (2009) 525–536.##]
An application of geometrical isometries in nonplanar molecules
2
2
In this paper we introduce a novel methodology to transmit the origin to the center of a polygon in a molecule structure such that the special axis be perpendicular to the plane containing the polygon. The mathematical calculation are described completely and the algorithm will be showed as a computer program.
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255
261


A.
Rezaei
University of Kashan
University of Kashan
Iran
a_rezaei@kashanu.ac.ir


A.
ReisiVanani
University of Kashan
University of Kashan
Iran
areisi@kashanu.ac.ir


S.
Masoum
University of Kashan
University of Kashan
Iran
masoum@kashanu.ac.ir
frame
isometry
orthogonal transformation
polygon
Nonplanar polycyclic molecule
[1. W. E. Barth, R. G. Lawton, Dibenzo[ghi,mno]fluoranthene, J. Am. Chem. Soc. 88##(1966) 380–381.##2. F. Cataldo, A. Graovac, O. Ori, The Mathematics and Topology of Fullerenes, Springer, Verlag, Berlin, 2011.##3. Z. Chen, C. S. Wannere, C. Corminboeuf, R. Puchta, P. V. R. Schleyer, Nucleusindependent chemical shifts (NICS) as an aromaticity criterion, Chemical ReviewsColumbus 105 (2005) 3842–3888.##4. H. Fallah Bagher Shaidaei, C. S. Wannere, C. Corminboeuf, R. Puchta, P. V. R. Schleyer, Which NICS aromaticity index for planar rings is best?, Organic Letters 8 (2006) 863–866.##5. R. Millman and G. Parker, Elements of Differential Geometry, PrenticeHall, 1977.##6. B. O'Neill, Elementary Differential Geometry, 2nd ed. Academic Press, 1997.##7. A. Reisi−Vanani, A. A. Rezaei, Evaluation of the aromaticity of nonplanar and bowlshaped molecules by NICS criterion, J. Mol. Graph. Model. 61 (2015) 85–88.##8. A. A. Rezaei, Polygonal tiling of some surfaces containing fullerene molecules, Iranian J. Math. Chem. 5 (2) (2014) 2 99–105.##9. A. A. Rezaei, Tiling fullerene surface with heptagon and octagon, Fullerenes, Nanotubes, Carbon Nanostructures 23 (12) (2015) 1033–1036.##10. H. Sakurai, T. Daiko, T. Hirao, A synthesis of sumanene, a fullerene fragment, Science 301 (2003) 1878–1878.##]
On evdegree and vedegree topological indices
2
2
Recently two new degree concepts have been defined in graph theory: evdegree and vedegree. Also the evdegree and vedegree Zagreb and Randić indices have been defined very recently as parallel of the classical definitions of Zagreb and Randić indices. It was shown that evdegree and vedegree topological indices can be used as possible tools in QSPR researches . In this paper we define the vedegree and evdegree Narumi–Katayama indices, investigate the predicting power of these novel indices and extremal graphs with respect to these novel topological indices. Also we give some basic mathematical properties of evdegree and vedegree NarumiKatayama and Zagreb indices.
1

263
277


B.
Sahin
Faculty of Science, Selçuk University, Konya, Turkey
Faculty of Science, Selçuk University,
Turkey
shnbnymn25@gmail.com


S.
Ediz
Faculty of Education, Yuzuncu Yil University, Van, Turkey
Faculty of Education, Yuzuncu Yil University,
Turkey
suleymanediz@yyu.edu.tr
evdegree
vedegree
evdegree topological indices
vedegree topological indices
[1. M. Chellali, T.W. Haynes, S.T. Hedetniemi, T.M. Lewis, On vedegrees and evdegrees in graphs, Discrete Math. 340 (2017) 31−38.##2. S. Ediz, Predicting some physicochemical properties of octane isomers: A topological approach using evdegree and vedegree Zagreb indices, Int. J. Syst. Sci. Appl. Math. 2 (2017), 87−92.##3. I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total πelectron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535−538.##4. B. Horoldagva, K. Das, T. Selenge, Complete characterization of graphs for direct comparing Zagreb indices, Discrete Appl. Math. 215 (2016) 146−154.##5. A. Ali, Z. Raza, A. Bhatti, A note on the augmented Zagreb index of cacti with fixed number of vertices and cycles, Kuwait J. Sci. 43 (2016) 11−17.##6. S. Ediz, Reduced second Zagreb index of bicyclic graphs with pendent vertices, Le Mathematiche 71 (2016) 135−147.##7. M. Randić, Characterization of molecular branching, J. Am. Chem. Soc. 97 (1975) 6609−6615.##8. R. K. Kincaid, S. J. Kunkler, M. D. Lamar, D. J. Phillips, Algorithms and complexity results for findings graphs with extremal Randić index, Networks 67 (2016) 338−347.##9. A. Banerjee, R. Mehatari, An eigenvalue localization theorem for stochastic matrices and its application to Randić matrices, Linear Algebra Appl. 505 (2016) 85−96.##10. R. Gu, F. Huang, X. Li, Skew Randić matrix and skew Randić energy, Trans. Comb. 5 (2016) 1−14.##11. B. Furtula, I. Gutman, A forgotten topological index, J Math. Chem. 53 (2015) 1184−1190.##12. N. De, S. M. Abu Nayeem, Computing the Findex of nanostar dendrimers, Pacific Sci. Rev. A: Nat. Sci. Eng. 18 (2016) 14−21.##13. W. Gao, M. K. Siddiqui, M. Imran, M. K. Jamil, M. R. Farahani, Forgotten topological index of chemical structure in drugs, Saudi Pharm. J. 24 (2016) 258−264.##14. H. Narumi, M. Katayama, Simple topological index. a newly devised index characterizing the topological nature of structural isomers of saturated hydrocarbons, Mem. Fac. Engin. Hokkaido Univ. 16 (1984) 209−214.##15. I. Gutman, M. Ghorbani, Some properties of the NarumiKatayama index, Appl. Math. Lett. 25 (2012) 1435−1438.##16. D. J. Klein, V. R. Rosenfeld, The degree–product index of Narumi and Katayama, MATCH Commun. Math. Comput. Chem. 64 (2010) 607−618.##17. D. J. Klein, V. R. Rosenfeld, The Narumi–Katayama degree–product index and the degree–product polynomial, in: I. Gutman, B. Furtula (Eds.), Novel Molecular Structure Descriptors—Theory and Applications II, Univ. Kragujevac, Kragujevac, 2010, pp. 79−90.##18. R. Todeschini, V. Consonni, Molecular Descriptors for Chemoinformatics, Wiley– VCH, Weinheim, 2009, p. 868.##19. A. Zolfi, A. R. Ashrafi, Extremal properties of NarumiKatayama index of chemical trees, Kragujevac J. Sci. 35 (2013), 71−76.##20. K. C. Das, N. Akgüneş, M. Togan, A. Yurttaş, İ. N. Cangül, A. S. Çevik, On the first Zagreb index and multiplicative Zagreb coindices of graphs, An. St. Univ. Ovidius ConstantaSeria Matematica 24 (2016), 153−176.##21. M. Eliasi, I. Gutman, A. Iranmanesh, Multiplicative versions of first Zagreb index, MATCH Commun. Math. Comput. Chem. 68 (2012) 217−230.##22. B. Basavanagoud, S. Patil, Multiplicative Zagreb indices and coindices of some derived graphs, Opuscula Math. 36 (2016), 287−299.##]
The second geometricarithmetic index for trees and unicyclic graphs
2
2
Let $G$ be a finite and simple graph with edge set $E(G)$. The second geometricarithmetic index is defined as $GA_2(G)=sum_{uvin E(G)}frac{2sqrt{n_un_v}}{n_u+n_v}$, where $n_u$ denotes the number of vertices in $G$ lying closer to $u$ than to $v$. In this paper we find a sharp upper bound for $GA_2(T)$, where $T$ is tree, in terms of the order and maximum degree of the tree. We also find a sharp upper bound for $GA_2(G)$, where $G$ is a unicyclic graph, in terms of the order, maximum degree and girth of $G$. In addition, we characterize the trees and unicyclic graphs which achieve the upper bounds.
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279
287


N.
Dehgardi
Department of Mathematics and Computer Science, Sirjan University of Technology,
Sirjan, Iran
Department of Mathematics and Computer Science,
Iran
n.dehgardi@sirjantech.ac.ir


H.
Aram
Department of Mathematics,
Gareziaeddin Center, Khoy Branch, Islamic Azad University, Khoy, Iran
Department of Mathematics,
Gareziaeddin Center,
Iran
hamideh.aram@gmail.com


A.
Khodkar
Department of Mathematics, University of West Georgia, Carrollton GA 30082
Department of Mathematics, University of
USA
akhodkar@westga.edu
Second geometricarithmetic index
Trees
Unicyclic graphs
[1 K. Ch. Das, I. Gutman and B. Furtula, On second geometricarithmetic index of graphs, Iranian J. Math. Chem. 1 (2010), 17–28.##2 K. Ch. Das, I. Gutman and B. Furtula, On the first geometricarithmetic index of graphs, Discrete Appl. Math. 159 (2011), 2030–2037.##3 H. Hua, Trees with given diameter and minimum second geometricarithmetic index, J. Math. Chem. 64 (2010), 631–638.##4 A. Madanshekaf and M. Moradi, The first geometricarithmetic index of some nanostar dendrimers, Iranian J. Math. Chem. 5 (2014), 1–6.##5 G. H. FathTabar, B. Furtula and I. Gutman, A new geometricarithmetic index, J. Math. Chem. 47 (2010), 477–486.##6 D. Vukicevic and B. Furtula, Topological index based on the ratios of geometrical and arithmetical means of endvertex degrees of edges, J. Math. Chem. 4 (2009), 1369–1376.##7 Y. Yuan, B. Zhou and N. Trinajstic, On geometricarithmetic index, J. Math. Chem. 47 (2010), 833–841.##]
On the saturation number of graphs
2
2
Let $G=(V,E)$ be a simple connected graph. A matching $M$ in a graph $G$ is a collection of edges of $G$ such that no two edges from $M$ share a vertex. A matching $M$ is maximal if it cannot be extended to a larger matching in $G$. The cardinality of any smallest maximal matching in $G$ is the saturation number of $G$ and is denoted by $s(G)$. In this paper we study the saturation number of the corona product of two specific graphs. We also consider some graphs with certain constructions that are of importance in chemistry and study their saturation number.
1

289
299


S.
Alikhani
Yazd University, iran
Yazd University, iran
Iran
alikhani@yazd.ac.ir


N.
Soltani
Yazd University, Iran
Yazd University, Iran
Iran
neda_soltani@ymail.com
Maximal matching
Saturation number
corona
[1. S. Alikhani, S. Jahari, M. Mehryar and R. Hasni, Counting the number of dominating sets of cactus chains, Optoelectron. Adv. Mater. − Rapid Comm. 8 (9−10) (2014) 955−960.##2. V. Andova, F. Kardoš and R. Škrekovski, Sandwiching saturation number of fullerene graphs, arXiv:1405.2197 (2014).##3. S. J. Cyvin and I. Gutman, Kekulé Structures in Benzenoid Hydrocarbons, Vol. 46 Lecture Notes in Chemistry, Springer Science, Heidelberg, 1988.##4. E. Deutsch and S. Klavžar, Computing the Hosoya polynomial of graphs from primary subgraphs, MATCH Commun. Math. Comput. Chem. 70 (2013) 627−644.##5. T. Došlić, On lower bounds of number of perfect matchings in fullerene graphs, J. Math. Chem. 24 (1998) 359−364.##6. T. Došlić, Fullerene graphs with exponentially many perfect matchings, J. Math. Chem. 41 (2007) 183−192.##7. T. Došlić, Saturation number of fullerene graphs, J. Math. Chem. 43 (2008) 647−657.##8. T. Došlić and I. Zubac, Saturation number of benzenoid graphs, MATCH Commun. Math. Comput. Chem. 73 (2015) 491−500.##9. J. Edmonds, Paths, trees, and flowers, Canad. J. Math. 17 (1965) 449−467.##10. J. Faudree, R. J. Faudree, R. J. Gould and M. S. Jacobson, Saturation numbers for trees, Electron. J. Combin. 16 (2009).##11. A. Frendrup, B. Hartnell and P. D. Vestergaard, A note on equimatchable graphs, Australas. J. Combin. 46 (2010) 185−190.##12. F. Harary, Graph Theory, AddisonWesley, Reading, MA, 1994.##13. F. Kardoš, D. Král, J. Miškuf and J. S. Sereni, Fullerene graphs have exponentially many perfect matchings, J. Math. Chem. 46 (2009) 443−447.##14. M. Klažar, Twelve countings with rooted plane trees, European J. Combin. 18 (1997) 195−210.##15. L. Lovász and M. D. Plummer, Matching Theory, Annals of Discrete Math. Vol. 29, NorthHolland, Amsterdam, 1986.##16. S. G. Wagner, On the number of matchings of a tree, European J. Combin. 28 (2007) 1322−1330.##17. D. B. West, Introduction to Graph Theory, Prentice Hall, Inc., Upper Saddle River, NJ, 1996.##18. H. Zhang and F. Zhang, New lower bound on the number of perfect matchings in fullerene graphs, J. Math. Chem. 30 (2001) 343−347.##]