Skip to main content
SpringerLink
Log in

Distance degree vector and scalar sequences of corona and lexicographic products of graphs with applications to dynamic NMR and dynamics of nonrigid molecules and proteins

  • Regular Article
  • Published:
Theoretical Chemistry Accounts Aims and scope Submit manuscript

Abstract

We have developed novel topological techniques to generate distance degree vector sequences (DDS) and distance degree scalar sequences (SS) of corona and lexicographic products of graphs both of which are of chemical and biochemical interest. Query ID="Q1" Text="Please confirm if the author names are presented accurately and in the correct sequence (given name, middle name/initial, family name). Author 1 Given name: [Medha Itagi] Last name [Huilgol]. Also, kindly confirm the details in the metadata are correct." We have obtained exact analytical expressions for such products for a variety of graphs and trees of chemical and biochemical interest. We have also outlined the applications of these graph products and their DDS to dynamic NMR spectroscopy and isomerization/conformational graphs of nonrigid molecules and proteins whose NMR and conformational dynamics graphs are expressible as lexicographic products of smaller graphs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Balasubramanian K (1980) Graph theoretical characterization of NMR Groups, nonrigid nuclear-spin species and the construction of symmetry adapted NMR spin functions. J Chem Phys 73:3321–3337

    Article  CAS  Google Scholar 

  2. Balasubramanian K (1995) Computer perception of NMR symmetry. J Magn Reson Ser A 112:182

    Article  CAS  Google Scholar 

  3. Balsubramanian K (1979) A generalized wreath product method for stereo and position isomers of polysubstituted organic compouds. Theor Chim Acta 51:37–54

    Article  Google Scholar 

  4. Balasubramanian K (1982) Symmetry groups of chemical graphs. Int J Quantum Chem 21:411–418

    Article  CAS  Google Scholar 

  5. Balasubramanian K (1988) Tree pruning method and lattice statistics on bethe lattices. J Math Chem 2:69–82

    Article  Google Scholar 

  6. Balasubramanian K (1982) Spectra of chemical trees. Int J Quantum Chem 21:581–590

    Article  CAS  Google Scholar 

  7. Balasubramanian K (2018) Relativity and the Jahn-Teller, Berry pseudorotations of TBP clusters: group theory, spin–orbit and combinatorial nuclear spin statistics of TBP Desargues-Levi isomerization graph. J Math Chem 56:2194–2225

    Article  CAS  Google Scholar 

  8. Wallace R (2012) Spontaneous symmetry breaking in a non-rigid molecule approach to intrinsically disordered proteins. Mol BioSyst 8:374–377

    Article  CAS  PubMed  Google Scholar 

  9. Wallace R (2017) Tools for the future: hidden symmetries. In: Computational psychiatry, pp 153–165. Springer, Cham

  10. Wallace R (2011) Multifunction moonlighting and intrinsically disordered proteins: information catalysis, non-rigid molecule symmetries and the ‘logic gate’ spectrum. C R Chim 14:1117–1121

    Article  CAS  Google Scholar 

  11. Balasubramanian K (2020) Nonrigid water octamer: computations with the 8-cube. J Comput Chem 41:2469–2484

    Article  CAS  PubMed  Google Scholar 

  12. Carbó-Dorca R (2020) Boolean hypercubes: the origin of a tagged recursive logic and the limits of artificial intelligence. Uni J Math Appl. https://doi.org/10.13140/RG.2.2.25069.13280

    Article  Google Scholar 

  13. Carbó-Dorca R, Chakraborty T (2019) Divagations about the periodic table: boolean hypercube and quantum similarity connections. J Comput Chem 40:2653–2663

    Article  PubMed  Google Scholar 

  14. Carbó-Dorca R (2018) Boolean hypercubes and the structure of Vector Spaces. J Math Sci Model 1:1–14

    Google Scholar 

  15. Carbó-Dorca R (2017) Natural vector spaces (inward power and Minkowski norm of a natural vector, natural boolean hypercubes) and a Fermat’s last theorem conjecture. J Math Chem 55:914–940

    Article  Google Scholar 

  16. Carbó-Dorca R (2020) Cantor-like transfinite sequences and Gödel-like incompleteness revealed by means of Mersenne transfinite dimensional boolean hypercube concatenation. J Math Chem 58:1–5. https://doi.org/10.1007/s10910-019-01075-4

    Article  CAS  Google Scholar 

  17. Carbó-Dorca R (2018) DNA unnatural base pairs and hypercubes. J Math Chem. 56:1353–1536. https://doi.org/10.1007/s10910-018-0866-9

    Article  CAS  Google Scholar 

  18. Carbó-Dorca R, Chakraborty T (2019) Hypercubes defined on n-ary sets, the Erdös–Faber–Lovász conjecture on graph coloring, and the description spaces of polypeptides and RNA. J Math Chem 57(10):2182–2194

    Article  Google Scholar 

  19. Mezey PG (1992) Similarity Analysis in two and three dimensions using lattice animals and ploycubes. J Math Chem 11:27–45

    Article  Google Scholar 

  20. Fralov A, Jako E, Mezey PG (2001) Logical models for molecular shapes and families. J Math Chem 30:389–409

    Article  Google Scholar 

  21. Mezey PG (2009) Some dimension problems in molecular databases. J Math Chem 45:1

    Article  CAS  Google Scholar 

  22. Mezey PG (1992) Shape similarity measures for molecular bodies: a three-dimensional topological approach in quantitative shape-activity relation. J Chem Inf Comput Sci 32:650

    Article  CAS  Google Scholar 

  23. Buckley F, Harary F (1990) Distance in graphs. Addison-Wesley

  24. Randić M (1979) Characterizations of atoms, molecules, and classes of molecules based on paths enumerations. MATCH 7:5–64

    Google Scholar 

  25. Kennedy JW, Quintas LV (1983) Extremal f-trees and embedding spaces for molecular graphs. Discrete Appl Math 5(2):191–209

    Article  Google Scholar 

  26. Bloom GS, Quintas LV, Kennedy JW (1981) Distance degree regular graphs, the theory and applications of graphs, 4th International conference, Western Michigan University, Kalamazoo, MI, May 1980. Wiley, New York, pp 95–108

    Google Scholar 

  27. Bloom GS, Kennedy JW, Quintas LV (1983) Some problems concerning distance and path degree sequences. Lect Notes Math 1018:179–190

    Article  Google Scholar 

  28. Gargano M, Quintas LV (1983) Smallest order pairs of non-isomorphic graphs having the same distance degree sequence and specified number of cycles, Notes from New York Graph Theory Day VI, pp 13–16, New York Academy of Sciences

  29. Bussemaker FC, Cobeljić S, Cvetković DB, Seidel JJ (1976) Computer investigation of cube Graphs, T. H. Report 76- WSK-01, Technological University Eindhoven, Eindhoven, The Netherlands

  30. Huilgol MI (2014) Distance degree regular graphs and distance degree injective graphs: an overview. J Discrete Math 2014:1–12

    Article  Google Scholar 

  31. Huilgol MI, Rajeshwari M, Syed Asif Ulla S (2013) Embedding in distance degree regular and distance degree injective graphs. Malaya J Math 4–1:134–141

    Google Scholar 

  32. Huilgol MI, Rajeshwari M, Syed Asif Ulla S (2012) Products of distance degree regular and distance degree injective graphs. J Discrete Math Sci Crypt 15–4(5):303–314

    Google Scholar 

  33. Huilgol MI, Walikar HB, Acharya BD (2011) On diameter three distance degree regular graphs. Adv Appl Discrete Math 7–1:39–61

    Google Scholar 

  34. Huilgol MI, Rajeshwari M, Syed Asif Ulla S (2011) Distance degree regular graphs and their eccentric digraphs. Int J Math Sci Eng Appl 5–6:405–416

    Google Scholar 

  35. Huilgol MI, Sriram V (2019) New results on distance degree sequences of graphs. Malaya J Math 7–2:345–352

    Article  Google Scholar 

  36. Quintas LV, Slater PJ (1981) Pairs of non-isomorphic graphs having the same path degree sequence. Match 12:75–86

    CAS  Google Scholar 

  37. Slater PJ (1982) Counter examples to Randić’s conjecture on distance degree sequences for trees. J Graph Theory 6–1:89–91

    Article  Google Scholar 

  38. Entringer RC, Jackson DE, Snyder DE (1976) Distance in graphs. Czech Math J 26:283–296

    Article  Google Scholar 

  39. Abiad A, Brimkovc B, Grigoriev A (2020) On the status sequences of trees, arXiv:1812.03765v2 [math.CO]

  40. Huilgol MI, Sriram V, Balasubramanian K (2020) Tensor and Cartesian Products for nanotori, nanotubes and zig-zag polyhex nanotubes and their applications to 13C NMR Spectroscopy. Mol Phys 117:1–24. https://doi.org/10.1080/00268976.2020.1817594

    Article  CAS  Google Scholar 

  41. Frucht R, Harary F (1970) On the corona of two graphs. Aequationes Math 4–3:322–325

    Article  Google Scholar 

  42. Adhikari B, Singh A, Yadav SK (2019) Corona product graphs with applications in signed networks. arXiv:1908.10018v1, [math.CO]

  43. Beineke LW, Harary F (1978) Consistent graphs with signed points. Rivista di mathematica per le scienze economiche e socials 1–2:81–88

    Google Scholar 

  44. Beineke LW, Harary F (1978) Consistency in marked graphs. J Math Psych 18–3:260–269

    Article  Google Scholar 

  45. Yero IG, Kuziak D, Rodriguez-Velazquez JA (2011) On the metric dimensional corona product graphs. Comput Math Appl 61:2793–2798

    Article  Google Scholar 

  46. De N (2007) Applications of corona product of graphs in computing topological indices of some special chemical graphs. Handbook of Research on Applied Cybernatics and Systems Science, 20 pages

  47. Feigenbaum J, Schäffer AA (1986) Recognizing composite graphs is equivalent to testing graph isomorphism. SIAM J Comput 15(2):619–627. https://doi.org/10.1137/0215045

    Article  Google Scholar 

  48. Imrich W, Klavžar S (2000) Product graphs: structure and recognition. Wiley, Hoboken

    Google Scholar 

  49. Cayley A (1881) On the analytical forms called trees. Am Math J 4:266–269

    Article  Google Scholar 

  50. Rains EM, Sloane NJ (1999) On Cayley's enumeration of alkanes (or 4-valent trees). J Integer Seq 2, Article 99.1.1

  51. Harary F, Robinson RW (1975) The number of achiral trees. J Reine Angew Math 278:322–335

    Google Scholar 

  52. Balasubramanian K, Gupta SP (2019) Quantum molecular dynamics, topological, group theoretical and graph theoretical studies of protein-protein interactions. Curr Top Med Chem 19:426–443

    Article  CAS  PubMed  Google Scholar 

  53. Arockiaraj M, Clement J, Tratnik N, Mushtaq S, Balasubramanian K (2020) Weighted Mostar indices as measures of molecular peripheral shapes with applications to graphene, graphyne and graphdiyne nanoribbons. SAR QSAR Environ Res 31:187–208

    Article  CAS  PubMed  Google Scholar 

  54. Arockiaraj M, Klavžar S, Clement J, Mushtaq S, Balasubramanian K (2019) Edge distance-based topological indices of strength-weighted graphs and their application to coronoid systems, carbon nanocones and SiO2 nanostructures. Mol Inform 38:1900039

    Article  CAS  Google Scholar 

  55. Car R, Parrinello M (1985) Unified approach for molecular dynamics and density-functional theory. Phys Rev Lett 55:2471–2474

    Article  CAS  PubMed  Google Scholar 

  56. Balasubramanian K (1997) Relativistic effects in chemistry: part A theory and techniques. Wiley, New York, p 301

    Google Scholar 

  57. Cao Z, Balasubramanian K, Calvert MG, Nitsche H (2009) Solvation effects on isomeric preferences of curium (III) complexes with multidentate phosphonopropionic acid ligands: CmH2PPA2+ and CmHPPA+ complexes. Inorg Chem 48(20):9700–9714

    Article  CAS  PubMed  Google Scholar 

  58. Balasubramanian K, Chaudhuri D (2008) Computational modeling of environmental plutonyl mono-, di-and tricarbonate complexes with Ca counterions: Structures and spectra: PuO2 (CO3)22-, PuO2 (CO3)2Ca, and PuO2 (CO3)3Ca3. Chem Phys Lett 450(4–6):196–202

    Article  CAS  Google Scholar 

  59. Balasubramanian K, Liao DW (1991) Spectroscopic constants and potential energy curves of Bi2 and Bi2-. J Chem Phy 95(5):3064–3073

  60. Balasubramanian K, Sumathi K, Dai D (1991) Group V trimers and their positive ions: The electronic structure and potential energy surfaces. J Chem Phy 95(5):3494–3505

  61. Balasubramanian K (1990) Electronic structure of (GaAs)2. Chem Phys Lett 171(1–2):58–62. https://doi.org/10.1016/0009-2614(90)80050-N

  62. Balasubramanian K (1989) Ten low‐lying electronic states of Pd3. The J Chem Phys 91(1):307–13. https://doi.org/10.1063/1.457518

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Bengaluru City University, Central College Campus, Bengaluru, 560 001, India

    Medha Itagi Huilgol & B. Divya

  2. School of Molecular Sciences, Arizona State University, Tempe, AZ, 85287-1604, USA

    Krishnan Balasubramanian

Authors
  1. Medha Itagi Huilgol
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. Divya
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Krishnan Balasubramanian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Krishnan Balasubramanian.

Additional information

Dedicated to Professor Ramon Carbó-Dorca on the occasion of his 80th Birthday.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Published as part of the special collection of articles "Festschrift in honour of Prof. Ramon Carbó-Dorca".

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huilgol, M.I., Divya, B. & Balasubramanian, K. Distance degree vector and scalar sequences of corona and lexicographic products of graphs with applications to dynamic NMR and dynamics of nonrigid molecules and proteins. Theor Chem Acc 140, 25 (2021). https://doi.org/10.1007/s00214-021-02719-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00214-021-02719-y

Keywords

Search

Navigation