Definition of Markov-Harary Invariants and Review of Classic Topological Indices and Databases in Biology, Parasitology, Technology,and Social-Legal Networks

Abstract

Graph and Complex Network theories are applied to different levels of matter organization such as genome networks, protein-protein networks, sexual disease transmission networks, linguistic networks, law and social networks, power electric networks or Internet. A very important fact is that we can use the numeric parameters called Topological Indices (TIs) to describe the connectivity patterns in all these kinds of Complex Networks despite the nature of the object they represent. The main reason for this success of TIs is the high flexibility of this theory to solve in a fast but rigorous way many apparently unrelated problems in all these disciplines. Another important reason for the success of TIs is that using these parameters as inputs we can find Quantitative Structure-Property Relationships (QSPR) models for any kind of bio-systems, at least in principle. In any case, there is a lack of manuscripts or issues focused on QSAR-like models with TIs and of networks more focused on Bioinformatics. In this sense, the present issue provides state-of-the-art reviews of some of these new computational approaches in this rapidly expanding area of Bioinformatics. Taking into account all the above-mentioned aspects, the present work intends to offer a common background to all the manuscripts presented in this special issue. In so doing, we make a review of classic TIs and Databases of Biology, Parasitology, Technology, and Social-Legal Networks. After that, we report a definition of a new class of TIs, coined here as Markov-Harary invariants. We also present the calculation of this class of TIs for different classes of networks. Next, we carry out a comparative study of these networks using the values of the new Markov-Harary TIs. Finally, we compare these new indices with another new class of TIs called Markov entropy values, which has been previously developed. © 2011 Bentham Science Publishers Ltd.