Topological and Thermodynamic Entropy Measures for COVID-19 Pandemic through Graph Theory
Abstract
:1. Introduction
2. Basic Concepts
3. Main Results for the Topological Indices
3.1. Topological Indices of Pandemic Trees
- (i)
- (ii)
- (iii)
- (iv)
- (v)
- (vi)
- (vii)
- (viii)
- (ix)
- ;
- (x)
- and whereand
- (i)
- (ii)
- (iii)
- ;
- ;
3.2. Topological Indices of Cayley Trees
- (i)
- (ii)
- (iii)
- (iv)
- (v)
- (vi)
- (vii)
- (viii)
- ;
- (ix)
- ;
- (x)
- and = where
- (i)
- (ii)
- (iii)
- :
3.3. Christmas Tree Network
- 1
- ST(2) is the complete graph K3 with its nodes labelled with u, l and r.
- 2
- The sth slim tree ST(s), with s ≥ 3, is composed of a root node u and two disjoint copies of (s − 1)th slim trees as the left subtree and right subtree, denoted byand, respectively and ST(s) = (V, E, u, l, r) is given by
- (i)
- (ii)
- (iii)
- (iv)
- (v)
- (vi)
- (vii)
- (viii)
- and where
- and .
- (i)
- (ii)
- 9;
- (iii)
- :
- and
3.4. Corona Product of Christmas Tree and a Path
- (i)
- (ii)
- (iii)
- (iv)
- (v)
- (vi)
- (vii)
- (viii)
- and ;
- and where
- and ;
- and where
- and
- and
- and
- and where
- and ;
- and where
- and
- and
- (i)
- (ii)
- (iii)
- ;
- :
- and ;
- and ;
- and
- and
- and .
4. Various Applications of Topological Indices for the COVID-19 Pandemic
- The difference in the awareness among all the countries and the general population on the severity of the disease and the necessary preventive actions that are needed as a result of internet and other forms of communications.
- The difference in the resources available for healthcare.
- The difference in number of qualified and trained virologists, doctors, nurses, government officials, government/private institutions, and other important frontline workers.
- The difference in information content available through research, innovations, and the data from the past pandemic, all of which can contribute to improved predictions. Researchers worldwide are working diligently to find a vaccine against the virus causing the COVID-19 pandemic. The WHO is working in collaboration with scientists, business, and global health organizations to accelerate the vaccine effectiveness and discovery [51].
- Other social, economic, and medical changes that occurred in the last 100 years.
5. Thermodynamic Entropy of Pandemic Trees
6. Stochasticity in Pandemic Tree Generation
- M1(T) = = 1957l2 – 15,660l + 33,270;
- (T) = = 1956l2 – 13,710l+25,440;
- ABC(T) =
- PI(T) = = 1957l + 3,816,150;
- Sz(T) == 531,706l + 15,153,306.
7. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
- Whitelaw, S.; A Mamas, M.; Topol, E.; Van Spall, H.G. Applications of digital technology in COVID-19 pandemic planning and response. Lancet Digit. Health 2020, 2, e435–e440. [Google Scholar] [CrossRef]
- Blackwood, J.C.; Childs, L.M. An introduction to compartmental modeling for the budding infectious disease modeler. Lett. Biomath. 2018, 5, 195–221. [Google Scholar] [CrossRef]
- Zhou, P.; Yang, X.-L.; Wang, X.-G.; Hu, B.; Zhang, L.; Zhang, W.; Si, H.-R.; Zhu, Y.; Li, B.; Huang, C.-L.; et al. A pneumonia outbreak associated with a new coronavirus of probable bat origin. Nature 2020, 579, 270–273. [Google Scholar] [CrossRef] [Green Version]
- Lai, C.-C.; Shih, T.-P.; Ko, W.-C.; Tang, H.-J.; Hsueh, P.-R. Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) and coronavirus disease-2019 (COVID-19): The epidemic and the challenges. Int. J. Antimicrob. Agents 2020, 55, 105924. [Google Scholar] [CrossRef]
- Andersen, K.G.; Rambaut, A.; Lipkin, W.I.; Holmes, E.C.; Garry, R.F. The proximal origin of SARS-CoV-2. Nat. Med. 2020, 26, 450–452. [Google Scholar] [CrossRef] [Green Version]
- Zhou, F.; Yu, T.; Du, R.; Fan, G.; Liu, Y.; Liu, Z.; Xiang, J.; Wang, Y.; Song, B.; Gu, X.; et al. Clinical course and risk factors for mortality of adult inpatients with COVID-19 in Wuhan, China: A retrospective cohort study. Lancet 2020, 395, 1054–1062. [Google Scholar] [CrossRef]
- Cui, J.; Li, F.; Shi, Z.-L. Origin and evolution of pathogenic coronaviruses. Nat. Rev. Genet. 2019, 17, 181–192. [Google Scholar] [CrossRef] [Green Version]
- Wu, J.T.; Leung, K.; Leung, G.M. Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: A modelling study. Lancet 2020, 395, 689–697. [Google Scholar] [CrossRef] [Green Version]
- Forster, P.; Forster, L.; Renfrew, C.; Forster, M. Phylogenetic network analysis of SARS-CoV-2 genomes. Proc. Natl. Acad. Sci. USA 2020, 117, 9241–9243. [Google Scholar] [CrossRef] [Green Version]
- Balasubramanian, K.; Khokhani, K.; Basak, S.C. Complex Graph Matrix Representations and Characterizations of Proteomic Maps and Chemically Induced Changes to Proteomes. J. Proteome Res. 2006, 5, 1133–1142. [Google Scholar] [CrossRef]
- Basak, S.C.; Grunwald, G.D.; Gute, B.D.; Balasubramanian, K.; Opitz, D. Use of statistical and neural net approaches in predicting toxicity of chemicals. J. Chem. Inf. Comput. Sci. 2000, 40, 885–890. [Google Scholar] [CrossRef]
- Matsen, F.A. Phylogenetics and the Human Microbiome. Syst. Biol. 2015, 64, e26–e41. [Google Scholar] [CrossRef]
- Delucchi, E. Nested set complexes of Dowling lattices and complexes of Dowling trees. J. Algebraic Comb. 2007, 26, 477–494. [Google Scholar] [CrossRef] [Green Version]
- Balasubramanian, K. Tree pruning and lattice statistics on Bethe lattices. J. Math. Chem. 1988, 2, 69–82. [Google Scholar] [CrossRef]
- Balasubramanian, K. Nested wreath groups and their applications to phylogeny in biology and Cayley trees in chemistry and physics. J. Math. Chem. 2017, 55, 195–222. [Google Scholar] [CrossRef]
- Balasubramanian, K. Spectra of chemical trees. Int. J. Quantum Chem. 1982, 21, 581–590. [Google Scholar] [CrossRef]
- Balasubramanian, K. Symmetry groups of chemical graphs. Int. J. Quantum Chem. 1982, 21, 411–418. [Google Scholar] [CrossRef]
- Sellers, P.H. On the Theory and Computation of Evolutionary Distances. SIAM J. Appl. Math. 1974, 26, 787–793. [Google Scholar] [CrossRef]
- Sellers, P.H. An algorithm for the distance between two finite sequences. J. Comb. Theory Ser. A 1974, 16, 253–258. [Google Scholar] [CrossRef] [Green Version]
- Fischer, M.; Klaere, S.; Nguyen, M.A.T.; Von Haeseler, A. On the group theoretical background of assigning stepwise mutations onto phylogenies. Algorithms Mol. Biol. 2012, 7, 36. [Google Scholar] [CrossRef] [Green Version]
- Yun, U.; Lee, G.; Kim, C.-H. The Smallest Valid Extension-Based Efficient, Rare Graph Pattern Mining, Considering Length-Decreasing Support Constraints and Symmetry Characteristics of Graphs. Symmetry 2016, 8, 32. [Google Scholar] [CrossRef] [Green Version]
- Xue, L.; Jing, S.; Miller, J.C.; Sun, W.; Li, H.; Estrada-Franco, J.G.; Hyman, J.M.; Zhu, H. A data-driven network model for the emerging COVID-19 epidemics in Wuhan, Toronto and Italy. Math. Biosci. 2020, 326, 108391. [Google Scholar] [CrossRef]
- Basavanagoud, B.; Desai, V.R.; Patil, S. (β,α)−Connectivity Index of Graphs. Appl. Math. Nonlinear Sci. 2017, 2, 21–30. [Google Scholar] [CrossRef] [Green Version]
- Gao, W.; Wang, W. The Vertex Version of Weighted Wiener Number for Bicyclic Molecular Structures. Comput. Math. Methods Med. 2015, 2015, 1–10. [Google Scholar] [CrossRef] [Green Version]
- Gao, W.; Farahani, M.R.; Shi, L. The forgotten topological index of some drug structures. Acta. Medica. Mediterr. 2016, 32, 579–585. [Google Scholar]
- Gao, W.; Wang, W.; Farahani, M.R. Topological Indices Study of Molecular Structure in Anticancer Drugs. J. Chem. 2016, 2016, 1–8. [Google Scholar] [CrossRef] [Green Version]
- Martínez-Pérez, Á.; Rodriguez, J.M. New Bounds for Topological Indices on Trees through Generalized Methods. Symmetry 2020, 12, 1097. [Google Scholar] [CrossRef]
- Atanasov, R.; Furtula, B.; Škrekovski, R. Trees with Minimum Weighted Szeged Index Are of a Large Diameter. Symmetry 2020, 12, 793. [Google Scholar] [CrossRef]
- Knor, M.; Imran, M.; Jamil, M.K.; Škrekovski, R. Remarks on Distance Based Topological Indices for ℓ-Apex Trees. Symmetry 2020, 12, 802. [Google Scholar] [CrossRef]
- Liu, J.-B.; Shaker, H.; Nadeem, I.; Farahani, M.R. Eccentric Connectivity Index of t-Polyacenic Nanotubes. Adv. Mater. Sci. Eng. 2019, 2019, 1–9. [Google Scholar] [CrossRef] [Green Version]
- Ghorbani, M.; Dehmer, M.; Emmert-Streib, F. Properties of Entropy-Based Topological Measures of Fullerenes. Mathematics 2020, 8, 740. [Google Scholar] [CrossRef]
- Ghorbani, M.; Khaki, A. A note on the fourth version of geometric-arithmetic index. Optoelectron. Adv. Mat. 2010, 4, 2212–2215. [Google Scholar]
- Gao, W.; Wu, H.; Siddiqui, M.K.; Baig, A.Q. Study of biological networks using graph theory. Saudi J. Biol. Sci. 2018, 25, 1212–1219. [Google Scholar] [CrossRef]
- Farahani, M.R.; Kanna, R.M.R. Fourth zagreb index of circumcoronene series of benzenoid. Leonardo Electron. J. Pract. Technol. 2015, 27, 155–161. [Google Scholar]
- Gutman, I.; Trinajstić, N. Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 1972, 17, 535–538. [Google Scholar] [CrossRef]
- Das, K.C.; Gutman, I.; Furtula, B. On atom-bond connectivity index. Chem. Phys. Lett. 2011, 511, 452–454. [Google Scholar] [CrossRef] [Green Version]
- Khadikar, P.V.; Karmarkar, S.; Agrawal, V.K. A Novel PI Index and Its Applications to QSPR/QSAR Studies. J. Chem. Inf. Comput. Sci. 2001, 41, 934–949. [Google Scholar] [CrossRef]
- Gutman, I. A formula for the Wiener number of trees and its extension to graphs containing cycles. Graph Theory Notes NY 1994, 27, 9–15. [Google Scholar]
- Song, C.; Hao, R.-X. Antimagic orientations for the complete k-ary trees. J. Comb. Optim. 2019, 38, 1077–1085. [Google Scholar] [CrossRef]
- Balasubramanian, K. Topo-Chemie-2020 is a package of codes that computes numerous degree-based, distance-based, eccentricity-based, neighbour-based topological indices, characteristic polynomials, matching polynomials, distance polynomials, distance degree vector sequences, walks and self-returning walks, and automorphisms of graphs.
- Weisstein, E.W. Cayley Tree, MathWorld-A Wolfram Web Resource. Available online: https://mathworld.wolfram.com/CayleyTree.html (accessed on 17 September 2020).
- Gutman, I.; Dobrynin, A.A. The Szeged index—A success story. Graph Theory Notes NY 1998, 34, 37–44. [Google Scholar]
- Hung, C.-N.; Hsu, L.-H.; Sung, T.-Y. Christmas tree: A versatile 1-fault-tolerant design for token rings. Inf. Process. Lett. 1999, 72, 55–63. [Google Scholar] [CrossRef]
- Nada, S.; Elrokh, A.; Elsakhawi, E.; Sabra, D. The corona between cycles and paths. J. Egypt. Math. Soc. 2017, 25, 111–118. [Google Scholar] [CrossRef]
- Khadikar, P.; Karmarkar, S.; Agrawal, V.; Singh, J.; Khadikar, P.V.; Lukovits, I.; Diudea, M.V. Szeged Index—Applications for Drug Modeling. Lett. Drug Des. Discov. 2005, 2, 606–624. [Google Scholar] [CrossRef] [Green Version]
- Basak, S.C.; Mills, D.; Mumtaz, M.M.; Balasubramanian, K. Use of topological indices in predicting aryl hydrocarbon receptor binding potency of dibenzofurans: A hierarchical QSAR approach. Indian J. Chem. 2003, 42, 1385–1391. [Google Scholar]
- Mondal, S.; De, N.; Pal, A. Topological Indices of Some Chemical Structures Applied for the Treatment of COVID-19 Patients. Polycycl. Aromat. Compd. 2020, 1–15. [Google Scholar] [CrossRef]
- Balasubramanian, K.; Gupta, S.P. Quantum Molecular Dynamics, Topological, Group Theoretical and Graph Theoretical Studies of Protein-Protein Interactions. Curr. Top. Med. Chem. 2019, 19, 426–443. [Google Scholar] [CrossRef]
- Balasubramanian, K. Mathematical and Computational Techniques for Drug Discovery: Promises and Developments. Curr. Top. Med. Chem. 2019, 18, 2774–2799. [Google Scholar] [CrossRef]
- Patil, V.M.; Narkhede, R.R.; Masand, N.; Rameshwar, S.; Cheke, R.S.; Balasubramanian, K. Molecular insights into Resveratrol and its analogs as SARS-CoV-2 (COVID-19) protease inhibitors. Coronaviruses 2020, in press. [Google Scholar]
- Available online: https://www.who.int/emergencies/diseases/novel-coronavirus-2019/covid-19-vaccines (accessed on 25 November 2020).
- Javelle, E.; Raoult, D. COVID-19 pandemic more than a century after the Spanish flu. Lancet Infect. Dis. 2020. [Google Scholar] [CrossRef]
- Dehmer, M.; Mowshowitz, A. A history of graph entropy measures. Inf. Sci. 2011, 181, 57–78. [Google Scholar] [CrossRef]
- Mowshowitz, A.; Dehmer, M. Entropy and the Complexity of Graphs Revisited. Entropy 2012, 14, 559–570. [Google Scholar] [CrossRef]
- Mowshowitz, A. Entropy and the complexity of graphs: I. An index of the relative complexity of a graph. Bull. Math. Biol. 1968, 30, 175–204. [Google Scholar] [CrossRef]
- Ghorbani, M.; Dehmer, M.; Rahmani, S.; Rajabi-Parsa, M. A Survey on Symmetry Group of Polyhedral Graphs. Symmetry 2020, 12, 370. [Google Scholar] [CrossRef] [Green Version]
Index | Dimension l, k | From Expressions 1 and 2 | Topo-Chemie-2020 [40] |
---|---|---|---|
GA4 | l = 3, k = 3 | 38.8016 | 38.801700143357856 |
l = 3, k = 4 | 83.5947 | 83.59488962413361 | |
l = 4, k = 3 | 119.6838 | 119.6839561076481 | |
l = 4, k = 4 | 339.1496 | 339.1498266447425 | |
Zg4 | l = 3, k = 3 | 399 | 399 |
l = 3, k = 4 | 876 | 876 | |
l = 4, k = 3 | 1692 | 1692 | |
l = 4, k = 4 | 4884 | 4884 | |
l = 3, k = 3 | 1.74212456 × 1039 | 1.742124563637115 × 1039 | |
l = 3, k = 4 | 1.98337192 × 1085 | 1.9833719240465464 × 1085 | |
l = 4, k = 3 | 3.75944893 × 10137 | 3.759448938495563 × 10137 | |
l = 4, k = 4 | Infinity | +Inf | |
Zg6 | l = 3, k = 3 | 1026 | 1026 |
l = 3, k = 4 | 2288 | 2288 | |
l = 4, k = 3 | 6000 | 6000 | |
l = 4, k = 4 | 17,584 | 17,584 | |
l = 3, k = 3 | 6.74664061 × 1054 | 6.746640616477462 × 1054 | |
l = 3, k = 4 | 4.66622007 × 10119 | 4.666220065428966 × 10119 | |
l = 4, k = 3 | 4.24758724 × 10202 | 4.247587242244764 × 10202 | |
l = 4, k = 4 | Infinity | +Inf | |
ABC5Π | l = 3, k = 3 | 2.0850970590 × 10−10 | 2.0850970590395853 × 10−10 |
l = 3, k = 4 | 7.2443933360 × 10−22 | 7.2443933361227945 × 10−22 | |
l = 4, k = 3 | 2.6373891592 × 10−37 | 2.6373891591075294 × 10−37 | |
l = 4, k = 4 | 1.26583295 × 10−105 | 1.2658329528339406 × 10−105 | |
l = 3, k = 3 | 1560 | 1560 | |
l = 3, k = 4 | 7140 | 7140 | |
l = 4, k = 3 | 14,520 | 14,520 | |
l = 4, k = 4 | 115,940 | 115,940 | |
l = 3, k = 3 | 3402 | 3402 | |
l = 3, k = 4 | 17,152 | 17,152 | |
l = 4, k = 3 | 44,712 | 44,712 | |
l = 4, k = 4 | 389,120 | 389,120 | |
l = 3, k = 3 | 228 | 228 | |
l = 3, k = 4 | 580 | 580 | |
l = 4, k = 3 | 714 | 714 | |
l = 4, k = 4 | 2372 | 2372 | |
l = 3, k = 3 | 288 | 288 | |
l = 3, k = 4 | 800 | 800 | |
l = 4, k = 3 | 936 | 936 | |
l = 4, k = 4 | 3360 | 3360 | |
l = 3, k = 3 | 30.8308 | 30.83052949654569 | |
l = 3, k = 4 | 68.6607 | 68.66073893642222 | |
l = 4, k = 3 | 94.13 | 94.12995706469174 | |
l = 4, k = 4 | 276.5956 | 276.59462680515827 |
Index | Dimension k, l | Expressions 3 and 4 | Topo-Chemie-2020 [40] |
---|---|---|---|
GA4 | k = 3, l = 3 | 20.8823882414 | 20.88238824139906 |
k = 3, l = 4 | 44.8676307844 | 44.86763078442802 | |
k = 4, l = 3 | 51.7356001911 | 51.73560019114378 | |
k = 4, l = 4 | 159.578608144 | 159.57860814353083 | |
Zg4 | k = 3, l = 3 | 207 | 207 |
k = 3, l = 4 | 609 | 609 | |
k = 4, l = 3 | 532 | 532 | |
k = 4, l = 4 | 2256 | 2256 | |
k = 3, l = 3 | 5.72086104 × 1020 | 5.7208610362887386 × 1020 | |
k = 3, l = 4 | 5.06517068 × 1050 | 5.065170680143067 × 1050 | |
k = 4, l = 3 | 2.09623003 × 1052 | 2.0962300336315637 × 1052 | |
k = 4, l = 4 | 2.71330692 × 10183 | 2.7133069091840815 × 10183 | |
Zg6 | k = 3, l = 3 | 516 | 516 |
k = 3, l = 4 | 2088 | 2088 | |
k = 4, l = 3 | 1368 | 1368 | |
k = 4, l = 4 | 8000 | 8000 | |
k = 3, l = 3 | 5.87731231 × 1028 | 5.877312307199999 × 1028 | |
k = 3, l = 4 | 1.5897237 × 1074 | 1.589723697730939 × 1074 | |
k = 4, l = 3 | 1.27482362 × 1073 | 1.2748236216396078 × 1073 | |
k = 4, l = 4 | 1.482028 × 10270 | 1.4820280048671849 × 10270 | |
ABC5Π | k = 3, l = 3 | 0.00000840649 | 0.000008406491899037542 |
k = 3, l = 4 | 4.4189082 × 10−14 | 4.419672639996076 × 10−14 | |
k = 4, l = 3 | 1.23642668 × 10−13 | 1.2364266794260217 × 10−13 | |
k = 4, l = 4 | 1.66132298 × 10−49 | 1.6913453312569212 × 10−49 | |
k = 3, l = 3 | 462 | 462 | |
k = 3, l = 4 | 2070 | 2070 | |
k = 4, l = 3 | 2756 | 2756 | |
k = 4, l = 4 | 25,760 | 25,760 | |
k = 3, l = 3 | 909 | 909 | |
k = 3, l = 4 | 5661 | 5661 | |
k = 4, l = 3 | 6304 | 6304 | |
k = 4, l = 4 | 82,336 | 82,336 | |
k = 3, l = 3 | 102 | 102 | |
k = 3, l = 4 | 222 | 222 | |
k = 4, l = 3 | 308 | 308 | |
k = 4, l = 4 | 956 | 956 | |
k = 3, l = 3 | 117 | 117 | |
k = 3, l = 4 | 261 | 261 | |
k = 4, l = 3 | 400 | 400 | |
k = 4, l = 4 | 1264 | 1264 | |
k = 3, l = 3 | 15.7979589714 | 15.797958971132715 | |
k = 3, l = 4 | 33.5959179422 | 33.59591794226542 | |
k = 4, l = 3 | 40.9748735074 | 40.974873507372514 | |
k = 4, l = 4 | 125.374110265 | 125.37411026490041 |
Index | Dimensions | From Expressions 5 and 6 | Topo-Chemie-2020 [40] |
---|---|---|---|
GA4 | s = 3 | 32.8823 | 32.882388241399056 |
s = 4 | 68.8671 | 68.86763078442803 | |
s = 5 | 140.8350 | 140.83501683597572 | |
Zg4 | s = 3 | 351 | 351 |
s = 4 | 993 | 993 | |
s = 5 | 2571 | 2571 | |
s = 3 | 5.10073079 × 1033 | 5.10077716500643 × 1033 | |
s = 4 | 4.01304165 × 1079 | 4.0130416580886176 × 1079 | |
s = 5 | 1.91126114 × 10177 | 1.9112611351919165 × 10177 | |
Zg6 | s = 3 | 948 | 948 |
s = 4 | 3624 | 3624 | |
s = 5 | 11,862 | 11,862 | |
s = 3 | 2.7848947 × 1047 | 2.7848946955924445 × 1047 | |
s = 4 | 3.54520226 × 10117 | 3.5452023119163584 × 10117 | |
s = 5 | 3.37859549 × 10269 | 3.3785954904846475 × 10269 | |
ABC5Π | s = 3 | 3.8618890813 × 10−9 | 3.861889064427591 × 10−9 |
s = 4 | 5.3059950832 × 10−22 | 5.305995082939164 × 10−22 | |
s = 5 | 1.7589082206 × 10−50 | 1.7589082206095107 × 10−50 | |
s = 3 | 198 | 198 | |
s = 4 | 414 | 414 | |
s = 5 | 846 | 846 | |
s = 3 | 297 | 297 | |
s = 4 | 621 | 621 | |
s = 5 | 1269 | 1269 | |
s = 3 | 22 | 22.000000000000004 | |
s = 4 | 46 | 45.99999999999997 | |
s = 5 | 94 | 94.00000000000011 |
Index | Dimension s, n | From Expressions 7 and 8 | Topo-Chemie-2020 [40] |
---|---|---|---|
GA4(CT(s)ʘPn) | s = 3, n = 3 | 142.7317 | 142.73175770071703 |
s = 4, n = 4 | 390.5798 | 390.5798602154981 | |
Zg4(CT(s)ʘPn) | s = 3, n = 3 | 1961 | 1961 |
s = 4, n = 4 | 6869 | 6869 | |
(CT(s)ʘPn) | s = 3, n = 3 | 9.65986697 × 10161 | 9.659866968127555 × 10161 |
s = 4, n = 4 | Infinity | +Inf | |
Zg6(CT(s)ʘPn) | s = 3, n = 3 | 6824 | 6824 |
s = 4, n = 4 | 30,567 | 30,567 | |
(CT(s)ʘPn) | s = 3, n = 3 | 4.3849717 × 10237 | 4.384971723725398 × 10237 |
s = 4, n = 4 | Infinity | +Inf | |
ABC5Π(CT(s)ʘPn) | s = 3,n = 3 | 1.4118921672 × 10−43 | 1.411892167070263 × 10−43 |
s = 4, n = 4 | 6.08702627 × 10−136 | 6.087026262790365 × 10−136 | |
s = 3, n = 3 | 1166 | 1166 | |
s = 4, n = 4 | 3450 | 3450 | |
s = 3, n = 3 | 2376 | 2376 | |
s = 4, n = 4 | 7567 | 7567 | |
s = 3, n = 3 | 93.33733429352 | 93.33733429351358 | |
s = 4, n = 4 | 251.70409168311 | 251.70409168311375 |
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Nandini, G.K.; Rajan, R.S.; Shantrinal, A.A.; Rajalaxmi, T.M.; Rajasingh, I.; Balasubramanian, K. Topological and Thermodynamic Entropy Measures for COVID-19 Pandemic through Graph Theory. Symmetry 2020, 12, 1992. https://doi.org/10.3390/sym12121992
Nandini GK, Rajan RS, Shantrinal AA, Rajalaxmi TM, Rajasingh I, Balasubramanian K. Topological and Thermodynamic Entropy Measures for COVID-19 Pandemic through Graph Theory. Symmetry. 2020; 12(12):1992. https://doi.org/10.3390/sym12121992
Chicago/Turabian StyleNandini, G. Kirithiga, R. Sundara Rajan, A. Arul Shantrinal, T. M. Rajalaxmi, Indra Rajasingh, and Krishnan Balasubramanian. 2020. "Topological and Thermodynamic Entropy Measures for COVID-19 Pandemic through Graph Theory" Symmetry 12, no. 12: 1992. https://doi.org/10.3390/sym12121992