Definition of the Subject and Its Importance
The concept of complexity has intrigued people from the beginning of history, but only in our times have attempts to quantify it begun to appear with a new science emerging – the science of complexity. Manifestation of complexity can be found everywhere in nature and life (Waldrop 1992), and different levels of complexity are encountered in arts, humanities, and sciences (Rouvray 2003).
Like many concepts in chemistry, the concept of complexity appears to be a fuzzy but useful concept (Rouvray 1997). The fuzziness of this concept has not prevented chemists from attempting to quantify it (Nikolić et al. 2003).
Here we are concerned with topological complexity of molecules. It should be stated at the outset that there are different levels of complexity regarding the structure of molecules (Bonchev and Seitz 1997), i.e., elemental or compositional or 1-dimensional complexity (which is determined by the partition of a graph’s vertices into...
Abbreviations
- Complexity (descriptor) index:
-
A complexity (descriptor) index is a number that is used to assess the complexity of a system.
- Complexity:
-
The word “complexity” is made up from the Latin roots “com” meaning “together” and “plectere” meaning “to plait.” Complexity is a difficult concept to define. One way to define it is as follows. A system is complex if it consists of a number of components interacting with each other in many different ways, so that these interactions sometimes lead to unexpected collective (emergent) properties. Hence, increasing complexity is associated with an increasing number of components in a system and with increasing versatility of their interactions.
- Graph:
-
A graph, sometimes called a nondirected graph and usually denoted by G, is a mathematical object which consists of two nonempty sets: one set, denoted usually by V, is a set of elements called vertices, and the other, usually denoted by E, is a set of unordered pairs of distinct elements of V called edges. Thus, G = (V,E). In directed graphs, E is a set of ordered pairs of elements of V. In multigraphs, more than one edge can join two vertices. A graph G is connected if every pair of vertices is joined by a path. If there is no path between two vertices in G, then G is the disconnected graph having two or more components. A graph G is a planar graph if it can be drawn in a plane in such a way that no two edges intersect. A dual G* of a planar graph G can be constructed in this way: Place a vertex in each region of G, including the exterior region, and if two regions have an edge e in common, join the corresponding vertices by an edge e* crossing only e. The inner dual is a subgraph of dual which does not contain the vertex corresponding to the exterior region. A graph is complete if each pair of its vertices is adjacent.
- Graph invariant:
-
An invariant of a graph G is a number associated with G which has the same value for any graph isomorphic to G.
- Graph-theoretical distance:
-
The length of the shortest path between two vertices in a graph G is the graph-theoretical distance.
- Laplacian matrix:
-
The Laplacian matrix L of a graph G is a real symmetric matrix whose diagonal elements are the vertex degrees of G and off-diagonal elements are −1 if vertices are connected in G, otherwise zero.
- Molecular graph:
-
Molecular graph is a 2-dimensional representation of a molecule and is generated by replacing atoms and bonds with vertices and edges, respectively. All molecular graphs in this article are connected graphs.
- Path:
-
A path is a sequence of edges, each edge sharing one vertex with the sequence-adjacent edges and sharing no vertices with any other edge. The length of a path is the number of edges it contains.
- Subgraph:
-
A subgraph of a graph G is a graph with all of its vertices and edges in G. A spanning subgraph is a subgraph containing all the vertices of G.
- Topological complexity:
-
Topological complexity is the complexity of graphs and is determined completely by the particular adjacency of a graph’s vertices – the issues of metrics and geometry are not relevant.
- Tree:
-
A tree is an acyclic graph – one which has no cycles. A spanning tree of a graph G is a connected, acyclic subgraph containing all the vertices of G.
- Vertex-adjacency matrix:
-
The vertex-adjacency matrix A is a 0–1 matrix representing graph. Entry 1 denotes adjacent vertices; all other entries are zero.
- Walk:
-
A walk in a graph G is an alternating sequence of vertices and edges of G, such that each edge e begins and ends with the vertices immediately preceding and following e in the sequence. The length of a walk is the number of edges it contains. The number of walks of length l beginning at vertex i and ending at vertex j is given by the i,j-element of the l-th power of the vertex-adjacency matrix, while the number of the self-returning walks of length l is given by the i,i-element of the l-th power of the vertex-adjacency matrix.
Bibliography
Primary Literature
Bertz SH (1982) Convergence, molecular complexity and synthetic analysis. J Am Chem Soc 104:5801–5803
Bertz SH (1983a) The first general index of molecular complexity. J Am Chem Soc 103:3599–3601
Bertz SH (1983b) A mathematical model of complexity. In: King RB (ed) Chemical applications of topology and graph theory. Elsevier, Amsterdam, pp 206–221
Bertz SH (1983c) On the complexity of graphs and molecules. Bull Math Biol 45:849–855
Bertz SH (2003) Complexity of molecules and their synthesis. In: Bonchev D, Rouvray DH (eds) Complexity in chemistry. Taylor & Francis, London, pp 91–156
Bertz SH, Rücker C (2004) In search of simplification: the use of topological complexity indices to guide retrosynthetic analysis. Croatica Chem Acta 77:221–235
Bertz SH, Sommer TJ (1997) Rigorous mathematical approaches to strategic bonds and synthetic analysis based on conceptually simple new complexity indices. Chem Commun 24:2409–2410
Bertz SH, Wright WF (1998) The graph theory approach to synthetic analysis: definition and application of molecular complexity and synthetic complexity. Graph Theory Notes N Y 35:32–48
Bertz SH, Zamfirescu C (2000) New complexity indices based on edge covers. MATCH Commun Math Comput Chem 42:39–70
Bonchev D (1990) Problems of computing molecular complexity. In: Rouvray DH (ed) Computational chemical graph theory. Nova Science Publishers, New York, pp 33–63
Bonchev D (1997) Novel indices for the topological complexity of molecules. SAR QSAR Environ Res 7:23–43
Bonchev D (2001) The overall wiener index – a new tool for characterizing molecular topology. J Chem Inf Comput Sci 41:582–592
Bonchev D, Buck GA (2007) From molecular to biological structure and back. J Chem Inf Model 47:909–917
Bonchev D, Polansky OE (1987) On the topological complexity of chemical systems. In: King RB, Rouvray DH (eds) Graph theory and topology in chemistry. Elsevier, Amsterdam, pp 125–158
Bonchev D, Seitz WA (1997) The concept of complexity in chemistry. In: Rouvray DH (ed) Concepts in chemistry: a contemporary challenge. Research Studies Press/Wiley, Taunton/New York, pp 353–381
Bonchev D, Trinajstić N (1977) Information theory, distance matrix and molecular branching. J Chem Phys 67:4517–4533
Bonchev D, Trinajstić N (2001) Overall molecular descriptors. 3. Overall Zagreb indices. Sar & QSAR Environ Res 12:213–235
Bonchev D, Kamenski D, Temkin ON (1987) Complexity index of the linear reaction mechanisms of chemical reactions. J Math Chem 1:345–388
Dancoff SM, Quastler H (1953) The information content and error rate in living things. In: Quastler H (ed) Essays on the use of information theory in biology. University of Illinois Press, Urbana
Fowler PW (2003) Complexity, spanning trees and relative energies of fullerene isomers. MATCH Commun Math Comput Chem 48:87–96
Gutman I, Trinajstić N (1972) Graph theory and molecular orbitals. III. Total π-electron energy of alternant hydrocarbons. Chem Phys Lett 17:535–538
Gutman I, Ruščić B, Trinajstić N, Wilcox CF Jr (1975) Graph theory and molecular orbitals. XII. Acyclic polyenes. J Chem Phys 62:3399–3405
Gutman I, Mallion RG, Essam JW (1983) Counting the spanning trees of a labelled molecular-graph. Mol Phys 50:859–877
Gutman I, Rücker C, Rücker G (2001) On walks in molecular graphs. J Chem Inf Comput Sci 41:739–745
Karreman G (1955) Topological information content and chemical reactions. Bull Math Biophys 17:279–285
Mihalić Z, Trinajstić N (1994) On the number of spanning trees in fullerenes. Fullerene Sci Technol 2:89–95
Minoli D (1975) Combinatorial graph complexity. Atti della Accademia Nazionale dei Lincei – Classe di Scienze fisiche, mathematiche e naturali (Serie 8) 59: 651–661
Mohar B (1989) Laplacian matrices of graphs. In: Graovac A (ed) MATH/CHEM/COMP 1989. Elsevier, Amsterdam, pp 1–8
Morgan HL (1965) The generation of a unique machine description for chemical structures – a technique developed at chemical abstracts service. J Chem Doc 5:107–113
Morowitz H (1955) Some order–disorder considerations in living systems. Bull Math Biophys 17:81–86
Mowshovitz A (1968) Entropy and the complexity of graphs: I. An index of the relative complexity of a graph. Bull Math Biophys 30:175–204
Nikolić S, Tolić IM, Trinajstić N, Baučić I (2000) On the Zagreb indices as complexity indices. Croatica Chemica Acta 73:909–921; see Note 28 and remarks by DJ Klein on complexity partial ordering
Nikolić S, Trinajstić N, Tolić IM, Rücker G, Rücker C (2003) On molecular complexity indices. In: Bonchev D, Rouvray DH (eds) Complexity in chemistry. Taylor & Francis, London, pp 29–89
Rajtmajer SM, Miličević A, Trinajstić N, Randić M, Vukičević D (2006) On the complexity of Archimedean solids. J Math Chem 39:119–132
Randić M (2001) On complexity of transitive graphs representing degenerate rearrangements. Croatica Chem Acta 74:683–705
Randić M, Plavšić D (2002) On the concept of molecular complexity. Croatica Chem Acta 75:107–116
Randić M, Plavšić D (2003) On characterization of molecular complexity. Int J Quant Chem 91:20–31
Rashevsky N (1955) Life, information theory and topology. Bull Math Biophys 17:229–235
Razinger M (1982) Extended connectivity of chemical graphs. Theor Chim Acta 61:581–586
Rouvray DH (1997) Are the concepts in chemistry all fuzzy? In: the concept of complexity in chemistry. In: Rouvray DH (ed) Concepts in chemistry: a contemporary challenge. Research Studies Press/Wiley, New York, pp 1–15
Rouvray DH (2003) An introduction to complexity. In: Bonchev D, Rouvray DH (eds) Complexity in chemistry. Taylor & Francis, London, pp 1–27
Rücker G, Rücker C (1990) Computer perception of constitutional (topological) symmetry: TOPSYM, a fast algorithm for partitioning atoms and pairwise relations among atoms into equivalence classes. J Chem Inf Comput Sci 30:187–191
Rücker G, Rücker C (1991a) On using the adjacency matrix power method for perception of symmetry and for isomorphism testing of highly intricate graphs. J Chem Inf Comput Sci 31:123–126
Rücker G, Rücker C (1991b) Isocodal and isospectral points, edges and pairs in graphs and how to cope with them in computerized symmetry recognition. J Chem Inf Comput Sci 31:422–427
Rücker G, Rücker C (1993) Counts of all walks as atomic and molecular descriptors. J Chem Inf Comput Sci 33:683–695
Rücker G, Rücker C (2000) Walk counts, labirinthicity and complexity of acyclic and cyclic graphs and molecules. J Chem Inf Comput Sci 40:99–106
Rücker G, Rücker C (2001a) On finding non-isomorphic connected subgraphs and distinct molecular substructures. J Chem Inf Comput Sci 41:314–320; erratum 825
Rücker G, Rücker C (2001b) Substructure, subgraph and walk counts as measures of the complexity of graphs and molecules. J Chem Inf Comput Sci 41:1457–1462
Shannon C, Weaver W (1949) Mathematical theory of communications. University of Illinois Press, Urbana
Todeschini R, Consonni V (2000) Handbook of molecular descriptors. Wiley-VCH, Weinheim, p 300
Trinajstić N, Babić D, Plavšić D, Amić D, Mihalić Z (1994) The Laplacian matrix in chemistry. J Chem Inf Comput Sci 34:368–376
Trucco E (1956a) A note on the information content of graphs. Bull Math Biophys 18:129–135
Trucco E (1956b) On the information content of graphs: compound symbols. Different states for each point. Bull Math Biophys 18:237–253
Waldrop MM (1992) Complexity: the emerging science at the edge of order and chaos. Touchstone/Simon & Schuster, New York
Wiener H (1947) Structural determination of paraffin boiling points. J Am Chem Soc 69:17–20
Books and Reviews
Devillers J, Balaban AT (eds) (1999) Topological descriptors and related descriptors in QSAR and QSPR. Gordon & Breach, Amsterdam
Cvetković DM, Doob M, Sachs H (1995) Spectra of graphs – theory and application, 3rd revised & enlarged edn. Barth, Heidelberg
Harary F (1971) Graph theory, 2nd printing. Addison-Wesley, Reading
Janežič D, Miličević A, Nikolić S, Trinajstić N (2007) Graph-theoretical matrices in chemistry. University of Kragujevac, Kragujevac
Trinajstić N (1992) Chemical graph theory, 2nd edn. CRC, Boca Raton
Wilson RJ (1972) Introduction to graph theory. Oliver & Boyd, Oxford
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Janežič, D., Miličević, A., Nikolić, S., Trinajstić, N. (2014). Topological Complexity of Molecules. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-3-642-27737-5_554-2
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Topological Complexity of Molecules- Published:
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DOI: https://doi.org/10.1007/978-3-642-27737-5_554-3
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Topological Complexity of Molecules- Published:
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DOI: https://doi.org/10.1007/978-3-642-27737-5_554-2