Topological properties of Sierpinski network and its application

BACKGROUND

Sierpinski graphs S(n,k) are largely studied because of their fractal nature with applications in topology, chemistry, mathematics of Tower of Hanoi, and computer sciences. Applications of molecular structure descriptors are a standard procedure that are used to correlate the biological activity of molecules with their chemical structures and thus can be helpful in the field of pharmacology.

OBJECTIVE

The aim of this article is to establish analytically closed computing formulae for eccentricity-based descriptors of Sierpinski networks and their regularizations. These computing formulae are useful to determine a large number of properties like thermodynamic properties, physicochemical properties, chemical and biological activity of chemical graphs.

METHODS

At first, vertex sets have been partitioned on the basis of their degrees, eccentricities, and frequencies of occurrence. Then these partitions are used to compute the eccentricity-based indices with the aid of some combinatorics.

RESULTS

The total eccentric index and eccentric-connectivity index have been computed. We also compute some eccentricity-based Zagreb indices of the Sierpinski networks. Moreover, a comparison has also been presented in the form of graphs.

CONCLUSION

These computations will help the readers to estimate the thermodynamic properties, physicochemical properties of chemical structures, which are of fractal nature and can not be dealt with easily. A 3D graphical representation is also presented to understand the dynamics of the aforementioned topological descriptors.

Efficient enumeration of monocyclic chemical graphs with given path frequencies

Background

The enumeration of chemical graphs (molecular graphs) satisfying given constraints is one of the fundamental problems in chemoinformatics and bioinformatics because it leads to a variety of useful applications including structure determination and development of novel chemical compounds.

Results

We consider the problem of enumerating chemical graphs with monocyclic structure (a graph structure that contains exactly one cycle) from a given set of feature vectors, where a feature vector represents the frequency of the prescribed paths in a chemical compound to be constructed and the set is specified by a pair of upper and lower feature vectors. To enumerate all tree-like (acyclic) chemical graphs from a given set of feature vectors, Shimizu et al. and Suzuki et al. proposed efficient branch-and-bound algorithms based on a fast tree enumeration algorithm. In this study, we devise a novel method for extending these algorithms to enumeration of chemical graphs with monocyclic structure by designing a fast algorithm for testing uniqueness. The results of computational experiments reveal that the computational efficiency of the new algorithm is as good as those for enumeration of tree-like chemical compounds.

Conclusions

We succeed in expanding the class of chemical graphs that are able to be enumerated efficiently.