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Topological Complexity of Molecules

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Encyclopedia of Complexity and Systems Science

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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...

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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.

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Authors and Affiliations

  1. University of Primorska, Faculty for Mathematics, Natural Sciences and Information Technologies, Koper, Slovenia

    Dušanka Janežič

  2. The Institute of Medical Research and Occupational Health, 291, HR-10002, Zagreb, Croatia

    Ante Miličević

  3. The Rugjer Bošković Institute, 180, HR-10002, Zagreb, Croatia

    Sonja Nikolić & Nenad Trinajstić

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  1. Dušanka Janežič
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  2. Ante Miličević
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  3. Sonja Nikolić
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  4. Nenad Trinajstić
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Correspondence to Dušanka Janežič .

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    Robert A. Meyers

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Janežič, D., Miličević, A., Nikolić, S., Trinajstić, N. (2018). Topological Complexity of Molecules. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27737-5_554-3

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  • DOI: https://doi.org/10.1007/978-3-642-27737-5_554-3

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Chapter history

  1. Latest

    Topological Complexity of Molecules
    Published:
    30 September 2017

    DOI: https://doi.org/10.1007/978-3-642-27737-5_554-3

  2. Original

    Topological Complexity of Molecules
    Published:
    06 June 2014

    DOI: https://doi.org/10.1007/978-3-642-27737-5_554-2

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