ORIGINAL_ARTICLE
A Novel Molecular Descriptor Derived from Weighted Line Graph
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.
https://ijmc.kashanu.ac.ir/article_93370_38c74963a18d68c1715935357d789545.pdf
2019-09-01T11:23:20
2020-08-07T11:23:20
195
207
10.22052/ijmc.2017.84168.1287
Weighted line graph
molecular descriptor
Physicochemical properties
Chandana
Adhikari
adhikarichandana@gmail.com
true
1
Sambalpur University
Sambalpur University
Sambalpur University
AUTHOR
Bijay
Mishra
bijaym@hotmail.com
true
2
School of Chemistry, Sambalpur University,
Jyoti Vihar - 768019
School of Chemistry, Sambalpur University,
Jyoti Vihar - 768019
School of Chemistry, Sambalpur University,
Jyoti Vihar - 768019
LEAD_AUTHOR
1. R. Todeschini and V, Consonni, Handbook of Molecular Dscriptors, Wiley VCH, Weinheim. 2000.
1
2. R. K. Mishra and B. K. Mishra,A critical assessment of closed- and open-shell heterocyclobutadienes, Chem. Phy. Lett. 151 (1988) 44−46.
2
3. F. Harary, Graph Theory, Addison Wesley Publishing Company, New York, 1969.
3
4. S. H. Bertz in: Chemical Applications of Topology and Graph Theory, R. B. King (Ed.), Elsevier, Amsterdam, 1993, p. 206.
4
5. S. H. Bertz, Branching in graphs and molecules, Discrete Appl. Math. 19 (1988) 65−83.
5
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.
6
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.
7
8. I. Gutman and Z. Tomovic, On the application of line graphs in quantitative structure-property studies, J. Serb. Chem. Soc. 65 (2000) 577−580.
8
9. I. Gutman, Z. Tomvic, B. K. Mishra and M. Kuanar, On the use of iterated line graphs in quantitative structure-property studies, Indian J. Chem. 40A (2001) 4−11.
9
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.
10
11. E. Estrada, Modelling the diamagnetic susceptibility of organic compounds by a sub-structural graph-theoretical approach, J. Chem. Soc. Faraday Trans. 94 (1998) 1407−1410.
11
12. E. Estrada, A computer-based 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.
12
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.
13
14. E. Estrada, Application of a novel graph-theoretic folding degree index to the study of steroid–DB3 antibody binding affinity, Comp. Biol. Chem. 27 (2003) 305−313.
14
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.
15
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.
16
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.
17
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.
18
19. B. Wu, Wiener index of line graphs, MATCH Commun. Math. Comput. Chem. 64 (2010) 699−706.
19
20. H. S. Ramane, I. Gutman and A. B. Ganagi, On diameter of line graphs, Iran. J. Math. Sci. Inf. 8 (2013) 105−109.
20
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.
21
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.
22
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.
23
24. A. A. Dobrynin, R. Entringer and I. Gutman, Wiener index of trees: theory and applications, Acta Appl. Math. 66 (2001) 211−249.
24
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.
25
26. M. Knor, P. Potočnik and R. Škrekovski, The Wiener index in iterated line graphs, Discrete Appl. Math. 160 (2012) 2234−2245.
26
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.
27
28. P. Dankelmann, I. Gutman, S. Mukwembi and H. C. Swart, The edge-Wiener index of a graph, Discrete Math. 309 (2009) 3452−3457.
28
29. S. M. Free and J. W. Wilson, A mathematical contribution to structure-activity studies, J. Med. Chem. 7 (1964) 395−399.
29
30. D. R. Lide, CRC Handbook of Chemistry and Physics, 73rd Ed., CRC Press, Boca Raton, FL, 1992.
30
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.
31
ORIGINAL_ARTICLE
Some Topological Indices of Edge Corona of Two Graphs
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 .
https://ijmc.kashanu.ac.ir/article_98970_6e2e05bf93818850a77f43d2cfeb1934.pdf
2019-09-01T11:23:20
2020-08-07T11:23:20
209
222
10.22052/ijmc.2017.34313.1132
Edge corona
Wiener index
Zagreb indices
Degree distance index
Gutman Index
Chandrashekar
Adiga
c_adiga@hotmail.com
true
1
University of Mysore, India
University of Mysore, India
University of Mysore, India
LEAD_AUTHOR
Malpashree
Raju
malpashree.5566@gmail.com
true
2
University of Mysore, India
University of Mysore, India
University of Mysore, India
AUTHOR
Rakshith
BIllava Ramanna
ranmsc08@yahoo.co.in
true
3
University of Mysore, India
University of Mysore, India
University of Mysore, India
AUTHOR
Anitha
Narasimhamurthy
nanitha@pes.edu
true
4
PES University, India
PES University, India
PES University, India
AUTHOR
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.
1
V. Andova, D. Dimitrov, J. Fink, R. Skrekovski, Bounds on Gutman index, MATCH Commun. Math. Comput. Chem.67 (2012) 515–524.
2
P. Dankelmann, I. Gutman, S. Mukwembi, H. C. Swart, On the degree distance of a graph, Discrete Appl. Math. 157 (2009) 2773–2777.
3
P. Dankelmann, I. Gutman, S. Mukwembi, H. C. Swart, The edge-Wiener index of a graph, Discrete Math. 309 (2009) 3452–3457.
4
A. A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: Theory and applications, Acta. Appl. Math. 66 (2001) 211–249.
5
A. A. Dobrynin, I. Gutman, S. Klav ar, P. igert, Wiener index of hexagonal systems, Acta Appl. Math. 72 (2002) 247–294.
6
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.
7
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.
8
R. Frucht, F. Harary, On the corona of two graphs, Aequationes Math. 4 (1970) 322–325.
9
L. Feng, W. Liu, The maximal Gutman index of bicyclic graphs, MATCH Commun. Math. Comput. Chem. 66 (2011) 699–708.
10
I. Gutman, Selected properties of Schultz molecular topological index, J. Chem. Inf. Coumput. Sci. 34 (1994) 1087–1089.
11
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.
12
I. Gutman, Degree–based topological indices, Croat. Chem. Acta 86 (2013) 351–361.
13
I. Gutman, K.C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004) 83–92.
14
I. Gutman, N. Trinajstić, Graph theory and molecular orbitals, Total electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538.
15
I. Gutman, S. Klav ar, B. Mohar (eds.), Fifty years of the Wiener index, MATCH Commun. Math. Comput. Chem. 35 (1997) 1–259.
16
Y. Hou, W.-C. Shiu, The spectrum of the edge corona of two graphs, Electron. J. Linear Algebra 20 (2010) 586–594.
17
A. Iranmanesh, I. Gutman, O. Khormali, A. Mahmiani, The edge versions of Wiener index, MATCH Commun. Math. Comput. Chem. 61 (2009) 663–672.
18
M. Khalifeh, H. Yousefi-Azari, A. R. Ashrafi, The first and second Zagreb indices of some graph operations, Discrete Appl. Math. 157 (2009) 804–811.
19
M. Liu, B. Liu, A survey on recent results of variable Wiener index, MATCH Commun. Math. Comput. Chem. 69 (2013) 491–520.
20
M. Knor, P. Potocnik, R. Skrekovski, Relationship between the edge-Wiener indexand the Gutman index of a graph, Discrete Appl. Math. 167 (2014) 197–201.
21
B. E. Sagan, Y. N. Yeh, P. Zhang, The Wiener polynomial of a graph, Int. J. Quant. Chem. 60 (1996) 959–969.
22
V. S. Agnes, Degree distance and Gutman index of corona product of graphs, Trans. Comb. 4 (3) (2015) 11–23.
23
I. Tomescu, Some extremal properties of the degree distance of a graph, Discrete Appl. Math. 98 (1999) 159–163.
24
A. I. Tomescu, Unicyclic and bicyclic graphs having minimum degree distance, Discrete Appl. Math. 156 (2008) 125–130.
25
H. Wiener, Structrual determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947) 17–20.
26
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.
27
ORIGINAL_ARTICLE
The distinguishing number and the distinguishing index of graphs from primary subgraphs
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 point-attaching 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.
https://ijmc.kashanu.ac.ir/article_101675_e08d1f5689447168e1f75217e9827f63.pdf
2019-09-01T11:23:20
2020-08-07T11:23:20
223
240
10.22052/ijmc.2019.152413.1400
Distinguishing index
distinguishing number
Chain
Link
Saeid
Alikhani
alikhani@yazd.ac.ir
true
1
Yazd University, Yazd, Iran
Yazd University, Yazd, Iran
Yazd University, Yazd, Iran
LEAD_AUTHOR
Samaneh
Soltani
s.soltani1979@gmail.com
true
2
Department of Mathematics, Yazd University, 89195-741, Yazd, Iran
Department of Mathematics, Yazd University, 89195-741, Yazd, Iran
Department of Mathematics, Yazd University, 89195-741, Yazd, Iran
AUTHOR
M.O. Albertson and K.L. Collins, Symmetry breaking in graphs, Electron. J. Combin. 3 (1996) #R18.
1
S. Alikhani and S. Soltani, Distinguishing number and distinguishing index of certain graphs, Filomat 31 (14) (2017) 4393–4404.
2
E. Deutsch and S. Klavžar, Computing Hosoya polynomials of graphs from primary subgraphs, MATCH Commun. Math. Comput. Chem. 70 (2013) 627–644.
3
M. V. Diudea, I. Gutman and J. Lorentz, Molecular Topology, Nova Science Publishers, Huntington, N.Y, 2001.
4
M. Ghorbani and M. A. Hosseinzadeh, On Wiener index of special case of link of fullerenes, Optoelectr. Adv. Mater. Rapid Comm. 4 (2010) 538–539.
5
M. Ghorbani and M. Songhori, Some topological indices of nanostar dendrimers, Iranian J. Math. Chem. 1 (2010) 57–65.
6
R. Hammack, W. Imrich and S. Klavžar, Handbook of product graphs (second edition), Taylor & Francis group, 2011.
7
R. Kalinowski and M. Pilśniak, Distinguishing graphs by edge colourings, European J. Combin. 45 (2015) 124–131.
8
T. Mansour and M. Schork, The PI index of bridge and chain graphs, MATCH Commun. Math. Comput. Chem. 61 (2009) 723–734.
9
T. Mansour and M. Schork, Wiener, hyper-Wiener, detour and hyper-detour indices of bridge and chain graphs, J. Math. Chem. 47 (2010) 72–98.
10
H. Wiener, Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69 (1947) 17–20.
11
ORIGINAL_ARTICLE
The Minimum Estrada Index of Spiro Compounds with k Quadrangles
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.
https://ijmc.kashanu.ac.ir/article_101892_5182c0c753fd3b745f298069defce2ad.pdf
2019-09-01T11:23:20
2020-08-07T11:23:20
241
249
10.22052/ijmc.2019.149094.1392
Strada index
spiro compound
point attaching graph
Mohammad
Iranmanesh
iranmanesh@yazd.ac.ir
true
1
Yazd University
Yazd University
Yazd University
LEAD_AUTHOR
Razieh
Nejati
nejati.razieh@gmail.com
true
2
Yazd University
Yazd University
Yazd University
AUTHOR
N. Biggs, Algebraic Graph Theory, Cambridge University Press, Cambridge, 1993.
1
G. Boros and V. H. Moll, A criterion for unimodality, Electron. J. Combin. 6 (1) (1999), #R10.
2
D. Cvetkovic, M. Doob and H. Sachs, Spectra of Graphs, Theory and Applications, Academic Press, New York, 1980.
3
H. Deng, A proof of a conjecture on the Estrada index, MATCH Commun. Math. Comput. Chem. 62 (3) (2009) 599–606.
4
E. Deutsch and S. Klavžar, Computing Hosoya polynomials of graphs from primary subgraphs, MATCH Commun. Math. Comput. Chem. 70 (2) (2013) 627–644.
5
E. Estrada, Characterization of 3D molecular structure, Chem. Phys. Lett. 319 (5−6) (2000) 713–718.
6
E. Estrada, Characterization of the folding degree of proteins, Bioinformatics 18 (5) (2002) 697–704.
7
I. Gutman and A. Graovac, Estrada index of cycles and paths, Chem. Phys. Lett. 436 (1−3) (2007) 294–296.
8
M. A. Iranmanesh and R. Nejati, On the Estrada index of point attaching strictk-quasi tree graphs, Kragujevac J. Math. 44 (2) (2020) 165–179.
9
J. Li, X. Li and L. Wang, The minimal Estrada index of trees with two maximum degree vertices, MATCH Commun. Math.Comput. Chem. 64 (2010) 799–810.
10
F. Li, L. Wei, J. Cao, F. Hu and H. Zhao, On the maximum Estrada index of 3-uniform linear hypertrees, Scientific World J. 8 (2014) 1–8.
11
F. Li, L. Wei, H. Zhao, F. Hu and X. Ma, On the Estrada index of cactus graphs, Discrete Appl. Math. 203 (2016) 94–105.
12
A. D. McNaught, Compendium of Chemical Terminology, Blackwell Science Publication, Oxford, 1997.
13
R. Rios, Enantioselective methodologies for the synthesis of spiro compounds, Chem. Soc. Rev. 41 (3) (2012) 1060–1074.
14
Y. Shang, Random lifts of graphs: network robustness based on the Estrada index, Appl. Math. E-Notes 12 (2012) 53–61.
15
Y. Shang, Biased edge failure in scale-free networks based on natural connectivity, Indian J. Phys. 86 (6) (2012) 485–488.
16
A. Von Zelewsky, Stereochemistry of Coordination Compounds, John Wiley & Sons, Chichester, 1996.
17
J. Zhang, B. Zhou and J. Li, On Estrada index of trees, Linear Algebra Appl. 434 (1) (2011) 215–223.
18
ORIGINAL_ARTICLE
An upwind local radial basis functions-finite difference (RBF-FD) method for solving compressible Euler equation with application in finite-rate Chemistry
The main aim of the current paper is to propose an upwind local radial basis functions-finite difference (RBF-FD) method for solving compressible Euler equation. The mathematical formulation of chemically reacting, inviscid, unsteady flows with species conservation equations and finite-rate 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 advection-diffusion 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.
https://ijmc.kashanu.ac.ir/article_102016_fe02c8a051aaaa92b3f3b7472e632fba.pdf
2019-09-01T11:23:20
2020-08-07T11:23:20
251
267
10.22052/ijmc.2017.106402.1325
Meshless Method
radial basis functions-finite difference (RBF-FD) technique
Compressible Euler equation
finite-rate Chemistry
Mostafa
Abbaszadeh
m.abbaszadeh@aut.ac.ir
true
1
Amirkabir University of Technology, Tehran, Iran, Faculty of Mathematics and Computer
Amirkabir University of Technology, Tehran, Iran, Faculty of Mathematics and Computer
Amirkabir University of Technology, Tehran, Iran, Faculty of Mathematics and Computer
LEAD_AUTHOR
Mehdi
Dehghan
mdehghan@aut.ac.ir
true
2
Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences,
Amirkabir University of Technology,
Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences,
Amirkabir University of Technology,
Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences,
Amirkabir University of Technology,
AUTHOR
Gholamreza
Karamali
gh_karamali@azad.ac.ir
true
3
Faculty of Basic Sciences, Shahid Sattari Aeronautical University of Sience and Technology,
South Mehrabad
Faculty of Basic Sciences, Shahid Sattari Aeronautical University of Sience and Technology,
South Mehrabad
Faculty of Basic Sciences, Shahid Sattari Aeronautical University of Sience and Technology,
South Mehrabad
AUTHOR
V. Bayona, M. Moscoso, M. Carretero and M. Kindelan, RBF-FD formulas and convergence properties, J. Comput. Phys. 229 (22) (2010) 8281–8295.
1
V. Bayona, M. Moscoso and M. Kindelan, Optimal constant shape parameter for multiquadric based RBF-FD method, J. Comput. Phys. 230 (19) (2011) 7384–7399.
2
V. Bayona, M. Moscoso and M. Kindelan, Gaussian RBF-FD weights and its corresponding local truncation errors, Eng. Anal. Bound. Elem. 36 (9) (2012) 1361–1369.
3
V. Bayona, M. Moscoso and M. Kindelan, Optimal variable shape parameter for multiquadric based RBF-FD method, J. Comput. Phys. 231 (6) (2012) 2466–2481.
4
R. Bellman, B. Kashef and J. Casti, Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, J. Comput. Phys.10 (1) (1972) 40–52.
5
E. F. Bollig, N. Flyer and G. Erlebacher, Solution to PDEs using radial basis function finite-differences (RBF-FD) on multiple GPUs, J. Comput. Phys. 231 (21) (2012) 7133–7151.
6
J.-C. Chassaing, X. Nogueira and S. Khelladi, Moving kriging reconstruction for high-order finite volume computation of compressible flows, Comput. Methods. Appl. Mech. Eng. 253 (2013) 463–478.
7
S. Chaturantabut, Dimension Reduction for Unsteady Nonlinear Partial Differential Equations via Empirical Interpolation Methods, MSc Thesis, Rice University, 2009.
8
S. Chaturantabut and D. C. Sorensen, A state space error estimate for POD-DEIM nonlinear model reduction, SIAM J. Numer. Anal. 50 (1) (2012) 46–63.
9
A. H. D. Cheng, Multiquadric and its shape parameter-A numerical investigation of error estimate, condition number, and round-of-error by arbitrary precision computation, Eng. Anal. Bound. Elem. 36 (2012) 220–239.
10
B. Dai, B. Zheng, Q. Liang and L. Wang, Numerical solution of transient heat conduction problems using improved meshless local Petrov-Galerkin method, Appl. Math. Comput. 219 (2013) 10044–10052.
11
L. Dawei, X. Xin, W. Zhi and C. Dehua, Investigation on the reynolds number simulation of supercritical airfoil, in: Digital Manufacturing and Automation (ICDMA), 2013 Fourth International Conference on, IEEE, 2013, pp. 7 40–744.
12
M. Dehghan and A. Nikpour, Numerical solution of the system of second-order boundary value problems using the local radial basis functions based differential quadrature collocation method, Appl. Math. Model. 37 (18) (2013) 8578–8599.
13
M. Dehghan and A. Nikpour, The solitary wave solution of coupled Klein–Gordon–Zakharov equations via two different numerical methods, Comput. Phys. Commun. 184 (9) (2013) 2145–2158.
14
T. A. Driscoll and B. Fornberg, Interpolation in the limit of increasingly flat radial basis functions, Comput. Math. Appl. 43 (3) (2002) 413–422.
15
J. Du, F. Fang, C. C. Pain, I. Navon, J. Zhu and D. A. Ham, Pod reduced-order unstructured mesh modeling applied to 2d and 3d fluid flow, Comput. Math. Appl. 65 (3) (2013) 362–379.
16
X. Du, C. Corre and A. Lerat, A third-order finite-volume residual-based scheme for the 2D Euler equations on unstructured grids, J. Comput. Phys. 230 (11) (2011) 4201–4215.
17
Y. T. Gu, Q. X. Wang and K. Y. Lam, A meshless local Kriging method for large deformation analyses, Comput. Methods Appl. Mech. Eng. 196 (2007) 1673–1684.
18
Y. T. Gu and G. R. Liu, A local point interpolation method for static and dynamic analysis of thin beams, Comput. Methods Appl. Mech. Eng. 190 (2001) 5515–5528.
19
Y. T. Gu, W. Wang, L. C. Zhang and X. Q. Feng, An enriched radial point interpolation method (e-RPIM) for analysis of crack tip fields, Eng. Fract. Mech. 78 (2011) 175–190.
20
F. Fang, C. Pain, I. Navon, G. Gorman, M. Piggott, P. Allison, P. Farrell and A. Goddard, A POD reduced order unstructured mesh ocean modelling method for moderate Reynolds number flows, Ocean Modelling 28 (1) (2009) 127–136.
21
G. E. Fasshauer, Meshfree Approximation Methods with MATLAB, World Scientific, Singapore, 2007.
22
N. Flyer, E. Lehto, S. Blaise, G. B. Wright and A. St-Cyr, A guide to RBF-generated finite differences for nonlinear transport: shallow water simulations on a sphere, J. Comput. Phys. 231 (11) (2012) 4078–4095.
23
B. Fornberg and E. Lehto, Stabilization of RBF-generated finite difference methods for convective PDEs, J. Comput. Phys. 230 (6) (2011) 2270–2285.
24
B. Fornberg, E. Lehto and C. Powell, Stable calculation of Gaussian-based RBF-FD stencils, Comput. Math. Appl. 65 (4) (2013) 627–637.
25
P. Gonzalez-Rodriguez, V. Bayona, M. Moscoso and M. Kindelan, Laurent series based RBF-FD method to avoid ill-conditioning, Eng. Anal. Bound Elem. 52 (2015) 24–31.
26
R. L. Hardy, Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res. 76 (1971) 1705–1915.
27
G. Hauke, D. Fuster and F. Lizarraga, Variational multiscale a posteriori error estimation for systems: The Euler and Navier–Stokes equations, Comput. Methods. Appl. Mech. Eng. 283 (2015) 1493–1524.
28
G. Hu, An adaptive finite volume method for 2D steady Euler equations with WENO reconstruction, J. Comput. Phys. 252 (2013) 591–605.
29
G. Hu, X. Meng and N. Yi, Adjoint-based an adaptive finite volume method for steady euler equations with non-oscillatory k-exact reconstruction, Comput. Fluids 139 (2016) 174–183.
30
G. Hu and N. Yi, An adaptive finite volume solver for steady Euler equations with non-oscillatory k-exact reconstruction, J. Comput. Phys. 312 (2016) 235–251.
31
H. Hu and O. A. Kandil, A hybrid boundary element–finite volume method for unsteady transonic airfoil flows, Eng. Anal. Bound Elem. 14 (2) (1994) 149–157.
32
S. Isaev, P. Baranov, I. Popov, A. Sudakov and A. Usachov, Improvement of aerodynamic characteristics of a thick airfoil with a vortex cell in sub-and transonic flow, Acta Astronautica 132 (2017) 204–220.
33
S. Jia, B. Yang, X. Zhao and J. Xu, Numerical simulation of far field acoustics of an airfoil using vortex method and 2-d fw-h equation, in: IOP Conf. Series: Materials Science and Engineering 52 (2013) 022047.
34
E. J. Kansa, Multiquadrics–A scattered data approximation scheme with applications to computational fluid-dynamics–II, Comput. Math. Appl. 19 (1990) 127–145.
35
E. J. Kansa, Multiquadrics A scattered data approximation scheme with applications to computational fluid dynamics - II, Comput. Math. Appl. 19 (1990) 147–161.
36
E. J. Kansa, R. C. Aldredge and Leevan Ling, Numerical simulation of two–dimensional combustion using mesh-free methods, Eng. Anal. Bound. Elem. 33 (2009) 940–950.
37
V. Kitsios, R. Kotapati, R. Mittal, A. Ooi, J. Soria and D. You, Numerical simulation of lift enhancement on a NACA 0015 airfoil using ZNMF jets, in: Proceedings of the Summer Program, Citeseer, 2006, p. 457.
38
S. S. Kutanaei, N. Roshan, A. Vosoughi, S. Saghafi, A. Barari and S. Soleimani, Numerical solution of Stokes flow in a circular cavity using mesh-free local RBF-DQ, Eng. Anal. Bound Elem. 36 (5) (2012) 633–638.
39
X. Li, Y. Liu, J. Kou and W. Zhang, Reduced-order thrust modeling for an efficiently flapping airfoil using system identification method, J. Fluids Struct. 69 (2017) 137–153.
40
G. R. Liu and Y. T. Gu, An Introduction to Meshfree Methods and Their Programming, Springer Dordrecht, Berlin, Heidelberg, New York, 2005.
41
Y. P. Marx, Numerical simulation of turbulent flows around airfoil and wing, in: Proceedings of the Eighth GAMM-Conference on Numerical Methods in Fluid Mechanics, Springer, 1990, pp. 323–332.
42
K. Nordanger, R. Holdahl, T. Kvamsdal, A. M. Kvarving and A. Rasheed, Simulation of airflow past a 2d NACA 0015 airfoil using an isogeometric incompressible navier–stokes solver with the spalart–allmaras turbulence model, Comput. Methods. Appl. Mech. Eng. 290 (2015) 183–208.
43
W. Ogana, Transonic integro-differential and integral equations with artificial viscosity, Eng. Anal. Bound Elem. 6 (3) (1989) 129–135.
44
R. Qin, L. Krivodonova, A discontinuous Galerkin method for solutions of the Euler equations on cartesian grids with embedded geometries, J. Comput. Sci. 4 (1-2) (2013) 24–35.
45
H. Rahimi, W. Medjroubi, B. Stoevesandt and J. Peinke, 2D numerical investigation of the laminar and turbulent flow over different airfoils using openfoam, in: J. Phys.: Conf. Ser. 555 (2014) 012070.
46
S. Ravindran, Reduced-order adaptive controllers for fluid flows using POD, J. Sci. Comput. 15 (4) (2000) 457–478.
47
S. S. Ravindran, A reduced-order approach for optimal control of fluids using proper orthogonal decomposition, Int. J. Numer. Methods Fluids 34 (5) (2000) 425–448.
48
P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys. 43 (2) (1981) 357–372.
49
A. Saadatmandi, N. Nafar and S. P. Toufighi, Numerical study on the reaction cum diffusion process in a spherical biocatalyst, Iranian J. Math. Chem. 5 (1) (2014) 47–61.
50
S. A. Sarra, A local radial basis function method for advection–diffusion–reaction equations on complexly shaped domains, Appl. Math. Comput. 218 (19) (2012) 9853–9865.
51
S. A. Sarra, Regularized symmetric positive definite matrix factorizations for linear systems arising from RBF interpolation and differentiation, Eng. Anal. Bound. Elem. 44 (2014) 76–86.
52
S. A. Sarra, Radial basis function approximation methods with extended precision floating point arithmetic, Eng. Anal. Bound. Elem. 35 (2011) 68–76.
53
T. K. Sengupta, A. Bhole and N. Sreejith, Direct numerical simulation of 2d transonic flows around airfoils, Comput. Fluids 88 (2013) 19–37.
54
V. Shankar, G. B. Wright, R. M. Kirby and A. L. Fogelson, A radial basis function (RBF)-finite difference (FD) method for diffusion and reaction–diffusion equations on surfaces, J. Sci. Comput. 63 (3) (2015) 745–768.
55
A. Shokri and M. Dehghan, Meshless method using radial basis functions for the numerical solution of two-dimensional complex Ginzburg-Landau equation, Comput. Model. Eng. Sci. 34 (2012) 333–358 .
56
C. Shu, Differential Quadrature and its Application in Engineering, Springer Science & Business Media, 2012.
57
C. Shu, H. Ding, H. Chen and T. Wang, An upwind local RBF-DQ method for simulation of inviscid compressible flows, Comput. Methods. Appl. Mech. Eng. 194 (18) (2005) 2001–2017.
58
C. Shu, H. Ding and K. Yeo, Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations, Comput. Methods. Appl. Mech. Eng. 192 (7) (2003) 941–954.
59
C. Shu, H. Ding and K. Yeo, Solution of partial differential equations by a global radial basis function-based differential quadrature method, Eng. Anal. Bound. Elem. 28 (10) (2004) 1217–1226.
60
R. tef nescu and I. M. Navon, POD/DEIM nonlinear model order reduction of an ADI implicit shallow water equations model, J. Comput. Phys. 237 (2013) 95–114.
61
A. I. Tolstykh, On using RBF-based differencing formulas for unstructured and mixed structured-unstructured grid calculations, in: Proceedings of the 16th IMACS World Congress on Scientific Computation, Applied Mathematics and Simulation, Lausanne, Switzerland, 2000, p. 6.
62
F. Tornabene, N. Fantuzzi, M. Bacciocchi, A. M. Neves and A. J. Ferreira, MLSDQ based on RBFs for the free vibrations of laminated composite doubly-curved shells, Composites Part B: Engineering 99 (2016) 30–47.
63
Q. Wang, Y.-X. Ren and W. Li, Compact high order finite volume method on unstructured grids ii: Extension to two-dimensional euler equations, J. Comput. Phys. 314 (2016) 883–908.
64
A.-M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Springer Science & Business Media, 2010.
65
H. Wendland, Scattered Data Approximation, in: Cambridge Monograph on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2005.
66
B. Xie, X. Deng, Z. Sun and F. Xiao, A hybrid pressure–density-based mach uniform algorithm for 2D Euler equations on unstructured grids by using multi-moment finite volume method, J. Comput. Phys. 335 (2017) 637–663.
67
L. W. Zhang, The IMLS-Ritz analysis of laminated CNT-reinforced composite quadrilateral plates subjected to a sudden transverse dynamic load, Composite Structures 180 (2017) 638–646.
68
G. Zhang, L. Ji and X. Hu, Vortex-induced vibration for an isolated circular cylinder under the wake interference of an oscillating airfoil: Part ii. single degree of freedom, Acta Astronautica 133 (2017) 311–323.
69
M. Zhao, M. Zhang and J. Xu, Numerical simulation of flow characteristics behind the aerodynamic performances on an airfoil with leading edge protuberances, Eng. Appl. Comput. Fluid Mech. 11 (1) (2017) 193–209.
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ORIGINAL_ARTICLE
Topological Efficiency of Some Product Graphs
The topological efficiency index of a connected graph $G,$ denoted by $\rho (G),$ is defined as $\rho(G)=\frac{2W(G)}{\left|V(G)\right|\underline w(G)},$ where $\underline w(G)=\text { min }\left\{w_v(G):v\in V(G)\right\}$ 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.
https://ijmc.kashanu.ac.ir/article_102017_57332d712f4df9e69475c3fdbbbe8a3c.pdf
2019-09-01T11:23:20
2020-08-07T11:23:20
269
278
10.22052/ijmc.2017.82177.1280
Wiener index
topological efficiency index
composite graph
Kannan
Pattabiraman
pramank@gmail.com
true
1
Annamalai University
Annamalai University
Annamalai University
LEAD_AUTHOR
Tholkappian
Suganya
suganyatpr@gmail.com
true
2
Annamalai University
Annamalai University
Annamalai University
AUTHOR
1. N. Alon and E. Lubetzky, Independent set in tensor graph powers, J. Graph
1
Theory 54 (2007) 73–87.
2
2. B. Bresar, W. Imrich, S. Klavžar and B. Zmazek, Hypercubes as direct products,
3
SIAM J. Discrete. Math. 18 (2005) 778–786.
4
3. S. Hossein-Zadeh, A. Iranmanesh, M. A. Hossein-Zadeh and A. R. Ashrafi,
5
Topological efficiency under graph operations, J. Appl. Math. Comput. 54 (2017)
6
69–80.
7
4. O. Ivanciuć, QSAR comparative study of Wiener descriptors for weighted
8
molecular graphs, J. Chem. Inf. Comput. Sci. 40 (2000) 1412–1422.
9
5. O. Ivanciuć, T. S. Balaban and A. T. Balaban, Reciprocal distance matrix, related
10
local vertex invariants and topological indices, J. Math. Chem. 12 (1993) 309–
11
6. W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition, Wiley,
12
New York, 2000.
13
7. F. Koorepazan-Moftakhar, A. R. Ashrafi, O. Ori and M. V. Putz, Topological
14
efficiency of fullerene, J. Comput. Theor. Nanosci. 12 (2015) 971–975.
15
8. F. Koorepazan-Moftakhar, A. R. Ashrafi, O. Ori and M. V. Putz, Topological
16
invariants of nanocones and fullerenes, Curr. Org. Chem. 19 (2015) 240–248.
17
9. H. Lei, T. Li, Y. Shi and H. Wang, Wiener polarity index and its generalization in
18
trees, MATCH Commun. Math. Comput. Chem. 78 (2017) 199–212.
19
10. S. Li and Y. Song, On the sum of all distances in bipartite graphs, Discrete Appl.
20
Math. 169 (2014) 176–185.
21
11. S. C. Li and W. Wei, Some edge-grafting transformation on the eccentricity
22
resistance-distance sum and their applications, Discrete Appl. Math. 211 (2016)
23
130–142.
24
12. J. Ma, Y. Shi, Z. Wang and J. Yue, On Wiener polarity index of bicyclic
25
networks, Sci. Rep. 6 (2016) 19066.
26
13. K. Pattabiraman and P. Paulraja, On some topological indices of the tensor
27
products of graphs, Discreate Appl. Math. 160 (2012) 267–79.
28
14. K. Pattabiraman and P. Paulraja, Wiener and vertex PI indices of the strong
29
product of graphs, Discuss. Math. Graph Theory 32 (2012) 749–769.
30
15. S. Sardana and A. K. Madan, Predicting anti-HIV activity of TIBO derivatives: A
31
computational approach using a novel topological descriptor, J. Mol. Model 8
32
(2002) 258–265.
33
16. D. Vukičević, F. Cataldo, O. Ori and A. Graovac, Topological efficiency of C66
34
fullerene, Chem. Phys. Lett. 501 (2011) 442–445.
35
17. H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc.
36
69 (1947) 17–20.
37
18. H. Zhang, S. Li and L. Zhao, On the further relation between the (revised) Szeged
38
index and the Wiener index of graphs, Discrete Appl. Math. 206 (2016) 152–164.
39