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A Novel Molecular Descriptor Derived from Weighted Line Graph
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2
The Bertz indices, derived by counting the number of connecting edges of line graphs of a molecule were used in deriving the QSPR models for the physicochemical properties of alkanes. The inability of these indices to identify the hetero centre in a chemical compound restricted their applications to hydrocarbons only. In the present work, a novel molecular descriptor has been derived from the weighted line graph of the molecular structure and applied in correlating the physicochemical properties of alkane isomers with these descriptors. A weight is tagged at the vertex of the line graph, which consequently modifies the weight of the edge. These descriptors were found to classify the alkane isomers and served well in deriving the QSPR models for various physicochemical properties. The mathematical calculations include the quantitative treatment on the role of substituents (alkyl) in governing the properties under study of the alkane isomers. Further, the use of weighted line graph in the enumeration of the topological index opens up a new vista on application to heteroatomic systems.
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195
207


Chandana
Adhikari
Sambalpur University
Sambalpur University
India
adhikarichandana@gmail.com


Bijay
Mishra
School of Chemistry, Sambalpur University,
Jyoti Vihar  768019
School of Chemistry, Sambalpur University,
Jyoti
India
bijaym@hotmail.com
Weighted line graph
molecular descriptor
Physicochemical properties
[1. R. Todeschini and V, Consonni, Handbook of Molecular Dscriptors, Wiley VCH, Weinheim. 2000.##2. R. K. Mishra and B. K. Mishra,A critical assessment of closed and openshell heterocyclobutadienes, Chem. Phy. Lett. 151 (1988) 44−46.##3. F. Harary, Graph Theory, Addison Wesley Publishing Company, New York, 1969.##4. S. H. Bertz in: Chemical Applications of Topology and Graph Theory, R. B. King (Ed.), Elsevier, Amsterdam, 1993, p. 206.##5. S. H. Bertz, Branching in graphs and molecules, Discrete Appl. Math. 19 (1988) 65−83.##6. I. Gutman, L. Popović, B. K. Mishra, M. Kuanar and E. Estrada, Application of line graphs in physical chemistry. Predicting the surface tensions of alkanes, J. Serb. Chem. Soc. 62 (1997) 1025−1030.##7. M. Kuanar, S. K. Kuanar, B. K. Mishra and I. Gutman, Correlation of line graph parameters with physicochemical properties of octane isomers, Indian J. Chem. 38A (1999) 525−528.##8. I. Gutman and Z. Tomovic, On the application of line graphs in quantitative structureproperty studies, J. Serb. Chem. Soc. 65 (2000) 577−580.##9. I. Gutman, Z. Tomvic, B. K. Mishra and M. Kuanar, On the use of iterated line graphs in quantitative structureproperty studies, Indian J. Chem. 40A (2001) 4−11.##10. Z. Tomovic and I. Gutman, Modeling boiling points of cycloalkanes by means of iterated line graph sequences, J. Chem. Inf. Comput. Sci. 41 (2001) 1041−1045.##11. E. Estrada, Modelling the diamagnetic susceptibility of organic compounds by a substructural graphtheoretical approach, J. Chem. Soc. Faraday Trans. 94 (1998) 1407−1410.##12. E. Estrada, A computerbased approach to describe the 13 C NMR chemical shifts of alkanes by the generalized spectral moments of iterated line graphs, Comput. Chem. 24 (2000) 193−201.##13. I. Gutman and E. Estrada, Topological indices based on the line graph of the molecular graph, J. Chem. Inf. Comput. Sci. 36 (1996) 541−543.##14. E. Estrada, Application of a novel graphtheoretic folding degree index to the study of steroid–DB3 antibody binding affinity, Comp. Biol. Chem. 27 (2003) 305−313.##15. N. Guevara, Fragmental graphs. A novel approach to generate a new family of descriptors. Applications to QSPR studies, J. Mol. Struct. (THEOCHEM) 493 (1999) 29−36.##16. A. A. Dobrynin and L. S. Mel’nikov, Wiener index for graphs and their line graphs with arbitrary large cyclomatic numbers, Appl. Math. Lett. 18 (2005) 307−312.##17. H. S. Ramane, K. P. Narayankar, S. S. Shirkol and A. B. Ganagi, Terminal Wiener index of line graphs, MATCH Commun. Math. Comput. Chem. 69 (2013) 775−782.##18. A. A. Dobrynin and L. S. Mel’nikov,Wiener index of generalized stars and their quadratic line graphs, Discuss. Math. Graph Theory 26 (2006) 161−175.##19. B. Wu, Wiener index of line graphs, MATCH Commun. Math. Comput. Chem. 64 (2010) 699−706.##20. H. S. Ramane, I. Gutman and A. B. Ganagi, On diameter of line graphs, Iran. J. Math. Sci. Inf. 8 (2013) 105−109.##21. A. A. Dobrynin and L. S. Mel'nikov, Trees and their quadratic line graphs having the same Wiener index, MATCH Commun. Math. Comput. Chem. 50 (2004) 145−164.##22. A. A. Dobrynin and L. S. Mel'nikov, Some results on the Wiener index of iterated line graphs, Electronic Notes Discrete Math. 22 (2005) 469−475.##23. M. Knor, P. Potočnik and R. Škrekovski, Wiener index of iterated line graphs of trees homeomorphic to H, Discrete Math. 313 (2013) 1104−1111.##24. A. A. Dobrynin, R. Entringer and I. Gutman, Wiener index of trees: theory and applications, Acta Appl. Math. 66 (2001) 211−249.##25. A. A. Dobrynin and L. S. Mel’nikov, Wiener index, line graphs and the cyclomatic number, MATCH Commun. Math. Comput. Chem. 53 (2005) 209−214.##26. M. Knor, P. Potočnik and R. Škrekovski, The Wiener index in iterated line graphs, Discrete Appl. Math. 160 (2012) 2234−2245.##27. M. Knor, P. Potočnik and R. Škrekovski, On a conjecture about Wiener index in iterated line graphs of trees, Discrete Math. 312 (2012) 1094−1105.##28. P. Dankelmann, I. Gutman, S. Mukwembi and H. C. Swart, The edgeWiener index of a graph, Discrete Math. 309 (2009) 3452−3457.##29. S. M. Free and J. W. Wilson, A mathematical contribution to structureactivity studies, J. Med. Chem. 7 (1964) 395−399.##30. D. R. Lide, CRC Handbook of Chemistry and Physics, 73rd Ed., CRC Press, Boca Raton, FL, 1992.##31. D. E. Needham, I. C. Wei and P. G. Seybold, Molecular modeling of the physical properties of alkanes, J. Am. Chem. Soc. 110 (1988) 4186−4194.##]
Some Topological Indices of Edge Corona of Two Graphs
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In this paper, we compute the Wiener index, first Zagreb index, second Zagreb index, degree distance index and Gutman index of edge corona of two graphs. Also in some cases we derive formulas for Weiner index, Zagreb indices, degree distance and Gutman index in terms of vertices and edges .
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209
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Chandrashekar
Adiga
University of Mysore, India
University of Mysore, India
India
c_adiga@hotmail.com


Malpashree
Raju
University of Mysore, India
University of Mysore, India
India
malpashree.5566@gmail.com


Rakshith
BIllava Ramanna
University of Mysore, India
University of Mysore, India
India
ranmsc08@yahoo.co.in


Anitha
Narasimhamurthy
PES University, India
PES University, India
India
nanitha@pes.edu
Edge corona
Wiener index
Zagreb indices
Degree distance index
Gutman Index
[A. R. Ashrafi, M. Ghorbani, M. Jalali, The vertex PI and Szeged indices of an infinite family of fullerenes, J. Theor. Comput. Chem.7 (2008) 221–231.##V. Andova, D. Dimitrov, J. Fink, R. Skrekovski, Bounds on Gutman index, MATCH Commun. Math. Comput. Chem.67 (2012) 515–524.##P. Dankelmann, I. Gutman, S. Mukwembi, H. C. Swart, On the degree distance of a graph, Discrete Appl. Math. 157 (2009) 2773–2777.##P. Dankelmann, I. Gutman, S. Mukwembi, H. C. Swart, The edgeWiener index of a graph, Discrete Math. 309 (2009) 3452–3457.##A. A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: Theory and applications, Acta. Appl. Math. 66 (2001) 211–249.##A. A. Dobrynin, I. Gutman, S. Klav ar, P. igert, Wiener index of hexagonal systems, Acta Appl. Math. 72 (2002) 247–294.##A. A. Dobrynin, A. A. Kochetova, Degree distance of a graph: a degree analogue of the Wiener index, J. Chem. Inf. Comput.Sci. 34 (1994) 1082–1086.##M. Essalih, M. E. Marraki and G. E. Hagri, Calculation of some topological indices of graphs, J. Theor. Appl. Inf. Tech. 30 (2011) 122–127.##R. Frucht, F. Harary, On the corona of two graphs, Aequationes Math. 4 (1970) 322–325.##L. Feng, W. Liu, The maximal Gutman index of bicyclic graphs, MATCH Commun. Math. Comput. Chem. 66 (2011) 699–708.##I. Gutman, Selected properties of Schultz molecular topological index, J. Chem. Inf. Coumput. Sci. 34 (1994) 1087–1089.##I. Gutman, A formula for the Wiener number of trees and its extension to graphs containing cycles, Graph Theory Notes New York 27 (1994) 9–15.##I. Gutman, Degree–based topological indices, Croat. Chem. Acta 86 (2013) 351–361.##I. Gutman, K.C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004) 83–92.##I. Gutman, N. Trinajstić, Graph theory and molecular orbitals, Total electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538.##I. Gutman, S. Klav ar, B. Mohar (eds.), Fifty years of the Wiener index, MATCH Commun. Math. Comput. Chem. 35 (1997) 1–259.##Y. Hou, W.C. Shiu, The spectrum of the edge corona of two graphs, Electron. J. Linear Algebra 20 (2010) 586–594.##A. Iranmanesh, I. Gutman, O. Khormali, A. Mahmiani, The edge versions of Wiener index, MATCH Commun. Math. Comput. Chem. 61 (2009) 663–672.##M. Khalifeh, H. YousefiAzari, A. R. Ashrafi, The first and second Zagreb indices of some graph operations, Discrete Appl. Math. 157 (2009) 804–811.##M. Liu, B. Liu, A survey on recent results of variable Wiener index, MATCH Commun. Math. Comput. Chem. 69 (2013) 491–520.##M. Knor, P. Potocnik, R. Skrekovski, Relationship between the edgeWiener indexand the Gutman index of a graph, Discrete Appl. Math. 167 (2014) 197–201.##B. E. Sagan, Y. N. Yeh, P. Zhang, The Wiener polynomial of a graph, Int. J. Quant. Chem. 60 (1996) 959–969.##V. S. Agnes, Degree distance and Gutman index of corona product of graphs, Trans. Comb. 4 (3) (2015) 11–23.##I. Tomescu, Some extremal properties of the degree distance of a graph, Discrete Appl. Math. 98 (1999) 159–163.##A. I. Tomescu, Unicyclic and bicyclic graphs having minimum degree distance, Discrete Appl. Math. 156 (2008) 125–130.##H. Wiener, Structrual determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947) 17–20.##Z. Yarahmadi, A. R. Ashrafi, The Szeged, vertex PI, first and second Zagreb indices of corona product of graphs, Filomat 26 (3) (2012) 467–472.##]
The distinguishing number and the distinguishing index of graphs from primary subgraphs
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2
The distinguishing number (index) D(G) (D'(G)) of a graph G is the least integer d such that G has an vertex labeling (edge labeling) with d labels that is preserved only by a trivial automorphism. Let G be a connected graph constructed from pairwise disjoint connected graphs G1,... ,Gk by selecting a vertex of G1, a vertex of G2, and identifying these two vertices. Then continue in this manner inductively. We say that G is obtained by pointattaching from G1, ... ,Gk and that Gi's are the primary subgraphs of G. In this paper, we consider some particular cases of these graphs that are of importance in chemistry and study their distinguishing number and distinguishing index.
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223
240


Saeid
Alikhani
Yazd University, Yazd, Iran
Yazd University, Yazd, Iran
I R Iran
alikhani@yazd.ac.ir


Samaneh
Soltani
Department of Mathematics, Yazd University, 89195741, Yazd, Iran
Department of Mathematics, Yazd University,
I R Iran
s.soltani1979@gmail.com
Distinguishing index
distinguishing number
Chain
Link
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The Minimum Estrada Index of Spiro Compounds with k Quadrangles
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2
Abstract. Let G = (V,E) be a finite and simple graph with λ1, λ2,...,λn as its eigenvalues.The Estrada index of G is EE(G) =∑ni=1e^{λi} . A spiro compound is a chemical compound that presents a twisted structure of two or more rings, in which 2 or 3 rings are linked together by one common atom. In this paper, we show that the symmetric and stable spiro compounds among all spiro compounds have the minimum Estrada index.
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241
249


Mohammad
Iranmanesh
Yazd University
Yazd University
I R Iran
iranmanesh@yazd.ac.ir


Razieh
Nejati
Yazd University
Yazd University
I R Iran
nejati.razieh@gmail.com
Strada index
spiro compound
point attaching graph
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An upwind local radial basis functionsfinite difference (RBFFD) method for solving compressible Euler equation with application in finiterate Chemistry
2
2
The main aim of the current paper is to propose an upwind local radial basis functionsfinite difference (RBFFD) method for solving compressible Euler equation. The mathematical formulation of chemically reacting, inviscid, unsteady flows with species conservation equations and finiterate chemistry is studied. The presented technique is based on the developed idea in [58]. For checking the ability of the new procedure, the compressible Euler equation is solved. This equation has been classified in category of system of advectiondiffusion equations. The solutions of advection equations have some shock, thus, special numerical methods should be applied for example discontinuous Galerkin and finite volume methods. Moreover, two problems are given that show the acceptable accuracy and efficiency of the proposed scheme.
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251
267


Mostafa
Abbaszadeh
Amirkabir University of Technology, Tehran, Iran, Faculty of Mathematics and Computer
Amirkabir University of Technology, Tehran,
I R Iran
m.abbaszadeh@aut.ac.ir


Mehdi
Dehghan
Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences,
Amirkabir University of Technology,
Department of Applied Mathematics, Faculty
Iran
mdehghan@aut.ac.ir


Gholamreza
Karamali
Faculty of Basic Sciences, Shahid Sattari Aeronautical University of Sience and Technology,
South Mehrabad
Faculty of Basic Sciences, Shahid Sattari
I R Iran
gh_karamali@azad.ac.ir
Meshless Method
radial basis functionsfinite difference (RBFFD) technique
Compressible Euler equation
finiterate Chemistry
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Topological Efficiency of Some Product Graphs
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The topological efficiency index of a connected graph $G,$ denoted by $rho (G),$ is defined as $rho(G)=frac{2W(G)}{leftV(G)rightunderline w(G)},$ where $underline w(G)=text { min }left{w_v(G):vin V(G)right}$ and $W(G)$ is the Wiener index of $G.$ In this paper, we obtain the value of topological efficiency index for some composite graphs such as tensor product, strong product, symmetric difference and disjunction of two connected graphs. Further, we have obtained the topological efficiency index for a double graph of a given graph.
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Kannan
Pattabiraman
Annamalai University
Annamalai University
India
pramank@gmail.com


Tholkappian
Suganya
Annamalai University
Annamalai University
India
suganyatpr@gmail.com
Wiener index
topological efficiency index
composite graph