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Complexity of molecules

Nikolić, Sonja; Trinajstić, Nenad
Complexity of molecules // Proceedings of the international conference of computational methods in sciences and engineering 2003 (ICCMSE 2003) / Simos, E. Theodore (ur.).
Singapore: World Scientific Publishing Co. Pte. Ltd. , Singapore, New York, London, 2003. str. 454-456

Complexity of molecules

Nikolić, Sonja ; Trinajstić, Nenad

Vrsta, podvrsta i kategorija rada
Poglavlja u knjigama, znanstveni

Proceedings of the international conference of computational methods in sciences and engineering 2003 (ICCMSE 2003)

Simos, E. Theodore

World Scientific Publishing Co. Pte. Ltd. , Singapore, New York, London



Raspon stranica


Ključne riječi
Complexity of molcules, molecular graphs, Zagreb complexity indices, topological complexity index

We studied complexity of several classes of molecular graphs: linear and branch trees, (poly)cycles and general graphs. Trees and (poly)cycles are simple graphs that are graphs without multiple edges and loops. General graphs are graphs with multiple edges and loops. A loop is an edge with both of its vertices identical. We used trees to represent alkanes, cycles to represent cycloalkanes, various polycyclic graphs to represent cage-structures and general graphs to represent heterosystems. We employed in our study the following complexity indices: Bonchev's indices, Zagreb complexity indices, spanning trees and total walk counts. We considered the following structural features: a graph (molecule) size in terms of number of vertices and edges, branching, cyclicity, the number of loops and multiple edges and symmetry. Bonchev proposed the topological complexity index TC to account for a complexity of the molecular graphs: (1) The first sum in the formula is over all connected subgraphs of a graph G, the second sum is over all vertices in the subgraph s and di is the vertex degree. The vertex degrees in subgraphs are taken as they are in G. If the vertex degrees are taken as they are in the respective isolated subgraphs s, than we have another Bonchev's complexity index denoted by TC1. Zagreb complexity indices TM1, TM1*, TM2, TM2* are based on the original Zagreb indices M1 and M2 and the above Bonchev's idea of using subgraphs. M1 and M2 indices are defined as: (2) (3) TM1 and TM2 are defined as: (4) (5) Vertex degrees entering TM1 and TM2 for a given subgraph s are taken as in a graph G. If vertex degrees are used as appearing in a subgraph s, than above formulas give TM1* and TM2* indices. A spanning tree of a graph G, t(G), is a connected acyclic subgraph containing all vertices in G. In the case of trees (alkanes), the spanning tree is identical to the tree itself: t(tree)=1 (6) In the case of monocycles (cycloalkanes) the number of spanning trees is equal to the cycle size: t(cycle)=V (7) where V is the number of the vertices in the cycle. There are a number of methods available to compute the number of spanning trees8. The total walk count twc is given as a sum of the molecular walk count mwc: (8) A walk in a (molecular) graph is an alternating sequence of vertices and edges. The length of the walk is the number of edges in it. The molecular walk count of length ℓ mwcℓ is given as a sum of all atomic walks of length ℓ (9) where awcℓ (i) is the atomic walk count of length ℓ of atom i. Formulas (8) and (9) apply equally to simple and general graphs. Our analysis shows: (i) all considered indices increased with the size and branching of alkanes ; (ii) all considered indices also increase with the size of the cycloalkanes (note that M1=M2 in the case of monocycles) ; (3) general graphs were studied only by twc. We found that increase in the number of loops and multiple edges considerable increases the value of twc ; (4) the increase in symmetry lowers the values of considered indices (twc index is not considered because it is symmetry independent). Additionally it should be emphasize that TC, TC1, TM1, TM1*, TM2, and TM2* indices depend on knowledge of all connected subgraphs. The number of subgraphs often enormously increases with the graph size. M1 and M2 are highly degenerate indices, so they cannot discriminate between many isomers. Spanning trees are useless in the case of acyclic structures. So our final conclusion is that all of the considered indices are useful to study some aspects of molecular complexity but we cannot say for any of them to be the best complexity index to use.

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Institut "Ruđer Bošković", Zagreb