Complexity of Molecules

Abstract

Several currently used measures of the complexity of molecules, such as, for example, those by Bertz and Randic or those based on the number of spanning trees, are briefly reviewed. We also proposed as complexity measures the sum of vertex-weights and the sum of edge-weights and their variants related to partition of vertex-weights and edge-weights into classes by their numerical values. The vertex-weights considered are the squares of the vertex-degrees, and the edge-weights are products of vertex-degrees making up the edges. Comparison is made between considered complexity measures for selected molecular graphs. All considered indices increase with increasing size and cyclicity and most with increasing branching. However, they differ regarding the influence of symmetry.

A history of graph entropy measures

“…Before starting, we emphasize that graph complexity can not be uniquely defined because there exist a lot of different structural features which contribute to the complexity of a graph. Similarly, to detect molecular complexity by taking the topological complexity of the underlying molecule into account is a challenging undertaking, see, e.g., [9,71]. Following Nikolić and Trinajstić [71], the topological complexity of a molecular graph is characterized by its number of vertices and edges, branching, cyclicity etc.…”

A Dual of the Cycle Theorem and its Application to Molecular Complexity

“…[7][8][9][10]) and the complexities of molecules (e.g., Refs. [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]), as well as in physical situations -e.g., electrical networks, such as in Ref. [27] and the papers that are cited therein.…”

“…[11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] Effectively, this Patency Index expresses a probability and, furthermore, we emphasise that it is analogous to the previously defined [25] Spanning-Tree Density.…”

“…In the context of chemical complexity [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] the reciprocal of STD(G) was suggested [25] as defining a quantitative measure of what Mallion & Trinajstić [25] called the Intricacy of G -the bigger the numerical value of this reciprocal the more intricate (in the sense of being 'complicated' or 'reticulated') the given graph, G, is.…”

“…[11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] Already, a referee has suggested that possible candidates for application of the Patency Index might be the objects studied by Diudea and coworkers (in, for example, Ref. [59]) as well as various geometrical objects that are constructed for nano-technology needs from strands of DNA or polypeptides, such as those studied in the work of Jerala et al (of which a representative example is Ref.…”