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2018 | OriginalPaper | Chapter

1. Basic Chemical Graph Theory

Author : Mircea Vasile Diudea

Published in: Multi-shell Polyhedral Clusters

Publisher: Springer International Publishing

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Abstract

Graph Theory applied in Chemistry is called Chemical Graph Theory. This interdisciplinary science takes problems (like isomer enumeration, structure elucidation, etc.) from Chemistry and solve them by Mathematics (using tools from Graph Theory, Set Theory or Combinatorics), thus influencing both Chemistry and Mathematics. This chapter introduces to basic definitions in Graph Theory: graph, walk, path, circuit, planar graph, graph invariant, vertex degree, chemical graph, etc. Then topological matrices are introduced: adjacency, distance, detour, combinatorial matrices, Wiener and Cluj matrices, walk matrix operator (combining three square matrices), reciprocal distance, and layer/shell matrices, on which the centrality indices are defined. Some info about topological symmetry is also presented.

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Amić D, Trinajstić N (1995) On the detour matrix. Croat Chem Acta 68:53–62

Balasubramanian K (1994) Computer generation of automorphism graphs of weighted graphs. J Chem Inf Comput Sci 34:1146–1150CrossRef

Balasubramanian K (1995a) Computer generation of nuclear equivalence classes based on the three-dimensional molecular structure. J Chem Inf Comput Sci 35:243–250CrossRef

Balasubramanian K (1995b) Computational strategies for the generation of equivalence classes of Hadamard matrices. J Chem Inf Comput Sci 35:581–589CrossRef

Balasubramanian K (1995c) Computer perception of molecular symmetry. J Chem Inf Comput Sci 35:761–770CrossRef

Bonchev D, Mekenyan O, Balaban AT (1989) Iterative procedure for the generalized graph center in polycyclic graphs. J Chem Inf Comput Sci 29:91–97CrossRef

Ciubotariu D (1987) Structură-reactivitate în clasa derivaţilor acidului carbonic. PhD Thesis, Timisoara, Romania

Ciubotariu D, Medeleanu M, Vlaia V, Olariu T, Ciubotariu C, Dragos D, Seiman C (2004) Molecular van der Waals space and topological indices from the distance matrix. Molecules 9:1053–1078CrossRef

Crippen GM (1977) A novel approach to calculation of conformation: distance geometry. J Comput Phys 24:96–107CrossRef

Diudea MV (1994) Layer matrices in molecular graphs. J Chem Inf Comput Sci 34:1064–1071CrossRef

Diudea MV (1996) Walk numbers ^eW_M: Wiener-type numbers of higher rank. J Chem Inf Comput Sci 36:535–540CrossRef

Diudea MV (1997a) Cluj matrix invariants. J Chem Inf Comput Sci 37:300–305CrossRef

Diudea MV (1997b) Cluj matrix, CJ_u: source of various graph descriptors. MATCH Commun Math Comput Chem 35:169–183

Diudea MV (1997c) Indices of reciprocal property or Harary indices. J Chem Inf Comput Sci 37:292–299CrossRef

Diudea MV (1999) Valencies of property. Croat Chem Acta 72:835–851

Diudea MV (2010) Nanomolecules and nanostructures—polynomials and indices, MCM, No. 10. University Kragujevac, Serbia

Diudea MV, Gutman I (1998) Wiener-type topological indices. Croat Chem Acta 71:21–51

Diudea MV, Randić M (1997) Matrix operator, W_(M1,M2,M3) and Schultz-type numbers. J Chem Inf Comput Sci 37:1095–1100CrossRef

Diudea MV, Ursu O (2003) Layer matrices and distance property descriptors. Indian J Chem 42A:1283–1294

Diudea MV, Topan MI, Graovac A (1994) Layer matrices of walk degrees. J Chem Inf Comput Sci 34:1071–1078

Diudea MV, Parv B, Gutman I (1997a) Detour-Cluj matrix and derived invariants. J Chem Inf Comput Sci 37:1101–1108CrossRef

Diudea M, Parv B, Topan MI (1997b) Derived Szeged and Cluj indices. J Serb Chem Soc 62:267–276

Diudea MV, Katona G, Lukovits I, Trinajstić N (1998) Detour and Cluj-detour indices. Croat Chem Acta 71:459–471

Diudea MV, Gutman I, Jäntschi L (2002) Molecular topology. NOVA, New York

Diudea MV, Florescu MS, Khadikar PV (2006) Molecular topology and its applications. EFICON, Bucharest

Dobrynin AA (1993) Degeneracy of some matrix invariant and derived topological indices. J Math Chem 14:175–184CrossRef

Dobrynin AA, Kochetova AA (1994) Degree distance of a graph: a degree analogue of the Wiener index. J Chem Inf Comput Sci 34:1082–1086CrossRef

Estrada E, Rodriguez L (1997) Matrix algebric manipulation of molecular graphs, Harary- and MTI-like molecular descriptors. MATCH Commun Math Comput Chem 35:157–167

Estrada E, Rodriguez L, Gutierrez A (1997) Matrix algebric manipulation of molecular graphs, distance and vertex-adjacency matrices. MATCH Commun Math Chem 35:145–156

Euler L (1752–1753) Elementa doctrinae solidorum. Novi Comm Acad Scient Imp Petrop 4:109–160

Graovac A, Babić D (1990) The evaluation of quantum chemical indices by the method of moments. Int J Quantum Chem Quantum Chem Symp 24:251–262CrossRef

Gutman I (1994) A formula for the Wiener number of trees and its extension to graphs containing cycles. Graph Theory Notes NY 27:9–15

Gutman I, Polansky OE (1986) Mathematical concepts in organic chemistry. Springer, Berlin, pp 108–116CrossRef

Halberstam FY, Quintas LV (1982) Distance and path degree sequences for cubic graphs. Pace University, New York. A note on table of distance and path degree sequences for cubic graphs, Pace University, New York

Harary F (1969) Graph theory. Addison-Wesley, ReadingCrossRef

Hargittai M, Hargittai I (2010) Symmetry through the eyes of a chemist. Springer, Dordrecht

Horn RA, Johnson CR (1985) Matrix analysis. Cambridge University Press, CambridgeCrossRef

Hungerford TW (1974) Algebra. Graduate texts in mathematics, vol 73. Springer, New York (Reprint of the original 1980)

Ionescu T (1973) Graphs-applications (in Romanian), Ed. Ped., Bucharest

Ivanciuc O, Balaban TS, Balaban AT (1993) Reciprocal distance matrix, related local vertex invariants and topological indices. J Math Chem 12:309–318CrossRef

Janežič D, Miličević A, Nikolić S, Trinajstić N (2007) Graph theoretical matrices in chemistry. Mathematical chemistry monographs. University Kragujevac, Kragujevac

Klein DJ, Lukovits I, Gutman I (1995) On the definition of the hyper-Wiener index for cycle-containing structures. J Chem Inf Comput Sci 35:50–52CrossRef

Kuratowski K (1930) Sur le Problème des Courbes Gauches en Topologie. Fund Math 15:271–283CrossRef

Lukovits I (1996) The detour index. Croat Chem Acta 69:873–882

Mirman R (1999) Point groups, space groups, crystals, molecules. World Scientific, River EdgeCrossRef

Petitjean M (2007) A definition of symmetry. Symmetry Cult Sci 18:99–119

Plavšić D, Nikolić S, Trinajstić N, Mihalić Z (1993) On the Harary index for the characterization of chemical graphs. J Math Chem 12:235–250CrossRef

Randić M (1991) Generalized molecular descriptors. J Math Chem 7:155–168CrossRef

Randić M (1993) Novel molecular description for structure-property studies. Chem Phys Lett 211:478–483CrossRef

Randić M (1995) Restricted random walks on graphs. Theor Chim Acta 92:97–106CrossRef

Randić M, Guo X, Oxley T, Krishnapriyan H (1993) Wiener matrix: source of novel graph invariants. J Chem Inf Comput Sci 33:700–716

Randić M, Guo X, Oxley T, Krishnapriyan H, Naylor L (1994) Wiener matrix invariants. J Chem Inf Comput Sci 34:361–367CrossRef

Razinger M, Balasubramanian K, Munk ME (1993) Graph automorphism perception algorithms in computer-enhanced structure elucidation. J Chem Inf Comput Sci 33:197–201CrossRef

Skorobogatov AV, Dobrynin AA (1988) Metric analysis of graphs. MATCH Commun Math Comput Chem 23:105–151

Sylvester JJ (1874) On an application of the new atomic theory to the graphical representation of the invariants and covariants of binary quantics—with three appendices. Am J Math 1:64–90CrossRef

Tratch SS, Stankevich MI, Zefirov NS (1990) Combinatorial models and algorithms in chemistry. The expanded Wiener number—a novel topological index. J Comput Chem 11:899–908CrossRef

Trinajstić N (1983) Chemical graph theory. CRC Press, Boca Raton

Ursu O, Diudea MV (2005) TOPOCLUJ software program. Babes-Bolyai University, Cluj

Wiener H (1947) Structural determination of paraffin boiling points. J Am Chem Soc 69:17–20CrossRef

Title: Basic Chemical Graph Theory
Author: Mircea Vasile Diudea
Publisher: Springer International Publishing
Book: Multi-shell Polyhedral Clusters
Print ISBN: 978-3-319-64121-8

Electronic ISBN: 978-3-319-64123-2

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-319-64123-2_1

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