COMPLEXITY OF MOLECULES
SONJA NIKOLIĆ and NENAD TRINAJSTIĆ
The Rugjer Bošković Institue, PP 180, 10002 Zagreb, Croatia
sonja@irb.hr; trina@irb.hr
We studied complexity of several classes of molecular graphs [1]:
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 [2] 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 [3], Zagreb complexity indices [4], spanning trees [2] and total
walk counts [5]. 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 [3]:
TC  
 d s
i
(1)
s verices
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:
M1 
d
2
i
(2)
vertex
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M
d d
i
(3)
j
edges
TM1 and TM2 are defined as:
TM 1  
 d s
2
i
(4)
s verices
TM 2  
s
 d d s 
i
j
(5)
edges
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 trees [8].
The total walk count twc is given as a sum of the molecular walk
count mwc [5-7]:
V -1
twc   mwc 
(8)
 1
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
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V
mwc    awc  (i)
(9)
i 1
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 [9].
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.
References
1. Trinajstić N., Chemical Graph Theory, 2nd revised edn. CRC Press,
Boca Raton, FL, 1992.
2. Harary F., Graph Theory, second printing. Addison-Wesley, Reading,
MA, 1972.
3. Bonchev D., Novel indices for the topological complexity of molecules.
SAR&QSAR Environ. Res. 7 (1997) pp. 23-43.
4. Nikolić S., Trinajstić N., Tolić I.M., Rücker G. and Rücker C., On
molecular complexity indices. In: Complexity in chemistry,
Introduction and fundamentals; Bonchev D. and Rouvray D.H., (eds.);
Mathematical chemistry series, Taylor&Francis, London, 2003.
5. Rücker G. and Rücker C., Counts of all walks as atomic and molecular
descriptors. J. Chem. Inf. Comput. Sci. 33 (1993) pp. 683-695.
6. Rücker G. and Rücker C., Walk counts, labyrinthicity and complexity
of acyclic and cyclic graphs and molecules. J. Chem. Inf. Comput. Sci.
40 (2000) pp. 99-106.
7. Rücker G. and Rücker C., On topological indices, boiling points and
cycloalkanes. J. Chem. Inf. Comput. Sci. 39 (1999) pp. 788-802.
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8. Nikolić S., Trinajstić N., Jurić A., Mihalić Z. and Krilov G.,
Complexity of some interesting (chemical) graphs. Croat. Chem. Acta
69 (1996) pp. 883-897.
9. Lukovits I., Miličević A., Nikolić S. and Trinajstić N., On walk counts
and complexity of general graphs. Internet Electron. J. Mol. Des. 2002,
1(8), pp. 388-400; http://www.biochempress.com
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