A Graph-Theoretical Approach to Calculate Vibrational Energies of Atomic and Subatomic Systems

Affiliation(s)

Department of Physical Chemistry, Faculty of Pharmacy, University of Valencia, Valencia, Spain.

Department of Physical Chemistry, Faculty of Pharmacy, University of Valencia, Valencia, Spain.

ABSTRACT

One of the challenges still pending in string theory and other particle physics related fields is the accurate prediction of the masses of the elementary particles defined in the standard model. In this paper an original algorithm to assign graphs to each of these particles is proposed. Based on this mapping, we demonstrate that certain indices associated with the topology of the graph (graph theoretical indices) are very effective in predicting the masses of the particles. Specifically, the spectral moments of the graph adjacency matrix weighted by edge degrees play a key role in the excellent correlations found. Moreover, the same topological pattern is found in other well known quantum systems such as the particle in a box and the vibrational frequencies of diatomic molecules, such as hydrogen. The results shown here open a suggestive pathway for the use of graph-theoretical approaches in predicting properties of elementary particles and other physical systems, which seem to match similar topological patterns.

One of the challenges still pending in string theory and other particle physics related fields is the accurate prediction of the masses of the elementary particles defined in the standard model. In this paper an original algorithm to assign graphs to each of these particles is proposed. Based on this mapping, we demonstrate that certain indices associated with the topology of the graph (graph theoretical indices) are very effective in predicting the masses of the particles. Specifically, the spectral moments of the graph adjacency matrix weighted by edge degrees play a key role in the excellent correlations found. Moreover, the same topological pattern is found in other well known quantum systems such as the particle in a box and the vibrational frequencies of diatomic molecules, such as hydrogen. The results shown here open a suggestive pathway for the use of graph-theoretical approaches in predicting properties of elementary particles and other physical systems, which seem to match similar topological patterns.

Cite this paper

J. Galvez, "A Graph-Theoretical Approach to Calculate Vibrational Energies of Atomic and Subatomic Systems,"*Open Journal of Physical Chemistry*, Vol. 2 No. 4, 2012, pp. 204-211. doi: 10.4236/ojpc.2012.24028.

J. Galvez, "A Graph-Theoretical Approach to Calculate Vibrational Energies of Atomic and Subatomic Systems,"

References

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[2] R. P. Feynman, “The Theory of Positrons,” Physical Review, Vol. 76, No. 6, 1949, pp. 749-759. doi:10.1103/PhysRev.76.749

[3] F. Harary, “Proof Techniques in Graph Theory,” Academic Press Inc., Burlington, 1969.

[4] M. C. Heydemann and B. Ducourthial, “Cayley Graphs and Interconnection Networks,” Physics, Science & Math, Vol. 497, No. 497, 1997, pp. 167-226.

[5] J. J. Sylvester and F. Franklin, “A Constructive Theory of Partitions, Arranged in Three Acts, an Interact and an Exodion,” American Journal of Mathematics, Vol. 5, No. 1, 1882, pp. 251-330. doi:10.2307/2369545

[6] T. M. Cover, “Comments on Broadcast Channels,” IEEE Transactions on Information Theory, Vol. 44, No. 6, 1998, pp. 2524-2530. doi:10.1109/18.720547

[7] N. Deo, “Graph Theory with Applications to Engineering and Computer Science,” PHI Learning Ltd., New Delhi, 2004.

[8] W. Ren, “Synchronization of Coupled Harmonic Oscillators with Local Interaction,” Automatica, Vol. 44, No. 12, 2008, pp. 3195-3200. doi:10.1016/j.automatica.2008.05.027

[9] A. W. Wolfe, “Social Network Analysis: Methods and Applications,” American Ethnologist, Vol. 24, No. 1, 1997, pp. 219-220. doi:10.1525/ae.1997.24.1.219

[10] F. Harary, “Graph Theory and Theoretical Physics,” Academic Press, New York, 1967.

[11] T. M. J. Fruchterman and E. M. Reingold, “Graph Drawing by Force-Directed Placement,” Software: Practice and Experience, Vol. 21, No. 11, 1991, pp. 1129-1164. doi:10.1002/spe.4380211102

[12] N. Cabibbo, “Unitary Symmetry and Leptonic Decays,” Physical Review Letters, Vol. 10, No. 12, 1963, pp. 531-533. doi:10.1103/PhysRevLett.10.531

[13] B. Greene, “The Elegant Universe,” Vintage Books, New York, 2000.

[14] R. Penrose, “Applications of Negative Dimensional Tensors,” Academic Press Inc., Burlington, 1971.

[15] J. Galvez, R. Garcia-Domenech and J. V. de Julian-Ortiz, “Assigning Wave Functions to Graphs: A Way to Introduce Novel Topological Indices,” Communications in Mathematical and in Computer Chemistry, Vol. 56, No. 3, 2006, pp. 509-518.

[16] I. Levine, “Quantum Chemistry,” 5th Edition, Prentice Hall, New Jersey ,1999.

[17] T. Filk, “Relational Interpretation of the Wave Function and a Possible Way Around Bell’s Theorem,” International Journal of Theoretical Physics, Vol. 45, No. 6, 2006, pp. 1166-1180. doi:10.1007/s10773-006-9125-0

[18] Dragon, “Talete Srl,” Milano, 2006. http://www.talete.mi.it/

[19] W. J. Dixon, M. B. Brown, L. Engelman and R. I. Jennrich, “7M Package,” University of California Press, San Francisco, 1990.

[20] E. Estrada, “Spectral Moments of the Edge Adjacency Matrix of Molecular Graphs. 1. Definition and Applications to the Prediction of Physical Properties of Alkanes,” Journal of Chemical Information and Computer Sciences, Vol. 36, No. 4, 1996, pp. 844-849. doi:10.1021/ci950187r

[21] G. N. Shah and T. A. Mir, “Are Elementary Particle Masses Related,” The 29th International Cosmic Ray Conference, Pune, 3-10 August 2005, pp. 219-222.

[22] S. Groote and J. G. Körner, “Spectral Moments of Two-Point Correlators in Perturbation Theory and Beyond,” Physical Review, Vol. 65, No. 3, 2002, 30 p. doi:10.1103/PhysRevD.65.036001

[23] B. Zhou, I. Gutman, J. A. de la Penña, J. Rada and L. Mendoza, “On Spectral Moments and Energy of Graphs,” Communications in Mathematical and in Computer Chemistry, Vol. 57, No. 1, 2007, pp 183-191.

[24] E. Estrada, “Quantum-Chemical Foundations of the Topological Substructure Molecular Design,” The Journal of Physical Chemistry, Vol. 112, No. 23, 2008, pp 5208-5217. doi:10.1021/jp8010712

[25] L. B. Kier, W. J. Murray, M. Randic and L. H. Hall, “Molecular Connectivity V: Connectivity Series Concept Applied to Density,” Journal of Pharmaceutical Sciences, Vol. 65, No. 8, 1976, pp. 1226-1230. doi:10.1002/jps.2600650824

[26] R. Garcia-Domenech, J. Galvez, J. V. de Julian-Ortiz and L. Pogliani, “Some New Trends in Chemical Graph Theory,” Chemical Reviews, Vol. 108, No. 3, 2008, pp. 1127-1169. doi:10.1021/cr0780006

[27] S. C. Basak, D. R. Mills, A. T. Balaban and B. D. Gute, “Prediction of Mutagenicity of Aromatic and Heteroaromatic Amines from Structure: A Hierarchical QSAR Approach,” Journal of Chemical Information and Computer Sciences, Vol. 41, No. 3, 2001, pp. 671-678. doi:10.1021/ci000126f

[28] P. Jasinski, B. Welsh, J. Galvez, D. Land, P. Zwolak, et al., “A Novel Quinoline, MT477: Suppresses Cell Signaling through Ras Molecular Pathway, Inhibits PKC Activity, and Demonstrates in vivo Anti-Tumor Activity against Human Carcinoma Cell Lines,” Investigational New Drugs, Vol. 26, No. 3, 2008, pp. 223-232. doi:10.1007/s10637-007-9096-x

[29] P. Jasinski, P. Zwolak, R. Isaksson, V. Bodempudi, K. Terai, et al., “MT103 Inhibits Tumor Growth with Minimal Toxicity in Murine Model of Lung Carcinoma via Induction of Apoptosis,” Investigational New Drugs, Vol. 29, No. 5, 2011, pp. 846-852. doi:10.1007/s10637-010-9432-4

[30] J. Galvez, J. Llompart, D. Land and G. M. Pasinetti, “Compositions for Treatment of Alzheimer’s Disease Using Abeta-Reducing and/or Abeta-Anti-Aggregation Compounds,” US Patent No. 2010114636, 2010.

[1] Smolin and J. Harnad, “The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next,” The Mathematical Intelligencer, Vol. 30, No. 3, 2008, pp. 66-69. doi:10.1007/BF02985383

[2] R. P. Feynman, “The Theory of Positrons,” Physical Review, Vol. 76, No. 6, 1949, pp. 749-759. doi:10.1103/PhysRev.76.749

[3] F. Harary, “Proof Techniques in Graph Theory,” Academic Press Inc., Burlington, 1969.

[4] M. C. Heydemann and B. Ducourthial, “Cayley Graphs and Interconnection Networks,” Physics, Science & Math, Vol. 497, No. 497, 1997, pp. 167-226.

[5] J. J. Sylvester and F. Franklin, “A Constructive Theory of Partitions, Arranged in Three Acts, an Interact and an Exodion,” American Journal of Mathematics, Vol. 5, No. 1, 1882, pp. 251-330. doi:10.2307/2369545

[6] T. M. Cover, “Comments on Broadcast Channels,” IEEE Transactions on Information Theory, Vol. 44, No. 6, 1998, pp. 2524-2530. doi:10.1109/18.720547

[7] N. Deo, “Graph Theory with Applications to Engineering and Computer Science,” PHI Learning Ltd., New Delhi, 2004.

[8] W. Ren, “Synchronization of Coupled Harmonic Oscillators with Local Interaction,” Automatica, Vol. 44, No. 12, 2008, pp. 3195-3200. doi:10.1016/j.automatica.2008.05.027

[9] A. W. Wolfe, “Social Network Analysis: Methods and Applications,” American Ethnologist, Vol. 24, No. 1, 1997, pp. 219-220. doi:10.1525/ae.1997.24.1.219

[10] F. Harary, “Graph Theory and Theoretical Physics,” Academic Press, New York, 1967.

[11] T. M. J. Fruchterman and E. M. Reingold, “Graph Drawing by Force-Directed Placement,” Software: Practice and Experience, Vol. 21, No. 11, 1991, pp. 1129-1164. doi:10.1002/spe.4380211102

[12] N. Cabibbo, “Unitary Symmetry and Leptonic Decays,” Physical Review Letters, Vol. 10, No. 12, 1963, pp. 531-533. doi:10.1103/PhysRevLett.10.531

[13] B. Greene, “The Elegant Universe,” Vintage Books, New York, 2000.

[14] R. Penrose, “Applications of Negative Dimensional Tensors,” Academic Press Inc., Burlington, 1971.

[15] J. Galvez, R. Garcia-Domenech and J. V. de Julian-Ortiz, “Assigning Wave Functions to Graphs: A Way to Introduce Novel Topological Indices,” Communications in Mathematical and in Computer Chemistry, Vol. 56, No. 3, 2006, pp. 509-518.

[16] I. Levine, “Quantum Chemistry,” 5th Edition, Prentice Hall, New Jersey ,1999.

[17] T. Filk, “Relational Interpretation of the Wave Function and a Possible Way Around Bell’s Theorem,” International Journal of Theoretical Physics, Vol. 45, No. 6, 2006, pp. 1166-1180. doi:10.1007/s10773-006-9125-0

[18] Dragon, “Talete Srl,” Milano, 2006. http://www.talete.mi.it/

[19] W. J. Dixon, M. B. Brown, L. Engelman and R. I. Jennrich, “7M Package,” University of California Press, San Francisco, 1990.

[20] E. Estrada, “Spectral Moments of the Edge Adjacency Matrix of Molecular Graphs. 1. Definition and Applications to the Prediction of Physical Properties of Alkanes,” Journal of Chemical Information and Computer Sciences, Vol. 36, No. 4, 1996, pp. 844-849. doi:10.1021/ci950187r

[21] G. N. Shah and T. A. Mir, “Are Elementary Particle Masses Related,” The 29th International Cosmic Ray Conference, Pune, 3-10 August 2005, pp. 219-222.

[22] S. Groote and J. G. Körner, “Spectral Moments of Two-Point Correlators in Perturbation Theory and Beyond,” Physical Review, Vol. 65, No. 3, 2002, 30 p. doi:10.1103/PhysRevD.65.036001

[23] B. Zhou, I. Gutman, J. A. de la Penña, J. Rada and L. Mendoza, “On Spectral Moments and Energy of Graphs,” Communications in Mathematical and in Computer Chemistry, Vol. 57, No. 1, 2007, pp 183-191.

[24] E. Estrada, “Quantum-Chemical Foundations of the Topological Substructure Molecular Design,” The Journal of Physical Chemistry, Vol. 112, No. 23, 2008, pp 5208-5217. doi:10.1021/jp8010712

[25] L. B. Kier, W. J. Murray, M. Randic and L. H. Hall, “Molecular Connectivity V: Connectivity Series Concept Applied to Density,” Journal of Pharmaceutical Sciences, Vol. 65, No. 8, 1976, pp. 1226-1230. doi:10.1002/jps.2600650824

[26] R. Garcia-Domenech, J. Galvez, J. V. de Julian-Ortiz and L. Pogliani, “Some New Trends in Chemical Graph Theory,” Chemical Reviews, Vol. 108, No. 3, 2008, pp. 1127-1169. doi:10.1021/cr0780006

[27] S. C. Basak, D. R. Mills, A. T. Balaban and B. D. Gute, “Prediction of Mutagenicity of Aromatic and Heteroaromatic Amines from Structure: A Hierarchical QSAR Approach,” Journal of Chemical Information and Computer Sciences, Vol. 41, No. 3, 2001, pp. 671-678. doi:10.1021/ci000126f

[28] P. Jasinski, B. Welsh, J. Galvez, D. Land, P. Zwolak, et al., “A Novel Quinoline, MT477: Suppresses Cell Signaling through Ras Molecular Pathway, Inhibits PKC Activity, and Demonstrates in vivo Anti-Tumor Activity against Human Carcinoma Cell Lines,” Investigational New Drugs, Vol. 26, No. 3, 2008, pp. 223-232. doi:10.1007/s10637-007-9096-x

[29] P. Jasinski, P. Zwolak, R. Isaksson, V. Bodempudi, K. Terai, et al., “MT103 Inhibits Tumor Growth with Minimal Toxicity in Murine Model of Lung Carcinoma via Induction of Apoptosis,” Investigational New Drugs, Vol. 29, No. 5, 2011, pp. 846-852. doi:10.1007/s10637-010-9432-4

[30] J. Galvez, J. Llompart, D. Land and G. M. Pasinetti, “Compositions for Treatment of Alzheimer’s Disease Using Abeta-Reducing and/or Abeta-Anti-Aggregation Compounds,” US Patent No. 2010114636, 2010.