Eccentric connectivity index has been found to have a low degeneracy and hence a significant potential of predicting biological activity of certain classes of chemical compounds. We present here explicit formulas for eccentric connectivity index of various families of graphs. We also show that the eccentric connectivity index grows at most polynomially with the number of vertices and determine the leading coefficient in the asymptotic behavior.

Nicotinic acetylcholine receptor (nAChR) genes form a highly conserved gene cluster at the lung cancer susceptibility locus 15q25.1. In this study, we show that the CHRNalpha3 gene encoding the nAChRalpha3 subunit is a frequent target of aberrant DNA hypermethylation and silencing in lung cancer, whereas the adjacent CHRNbeta4 and CHRNalpha5 genes exhibit moderate and no methylation, respectively. Treatment of cancer cells exhibiting CHRNalpha3 hypermethylation with DNA methylation inhibitors caused demethylation of the CHRNalpha3 promoter and gene reactivation. Restoring CHRNalpha3 levels through ectopic expression induced apoptotic cell death. Small hairpin RNA-mediated depletion of nAChRalpha3 in CHRNalpha3-expressing lung cancer cells elicited a dramatic Ca(2+) influx response in the presence of nicotine, followed by activation of the Akt survival pathway. CHRNalpha3-depleted cells were resistant to apoptosis-inducing agents, underscoring the importance of epigenetic silencing of the CHRNalpha3 gene in human cancer. In defining a mechanism of epigenetic control of nAChR expression in nonneuronal tissues, our findings offer a functional link between susceptibility locus 15q25.1 and lung cancer, and suggest nAChRs to be theranostic targets for cancer detection and chemoprevention.

This note introduces a new general conjecture correlating the dimensionality dT of an infinite lattice with N nodes to the asymptotic value of its Wiener Index W(N). In the limit of large N the general asymptotic behavior W(N)≈N s is proposed, where the exponent s and dT are related by the conjectured formula s=2+1/dT allowing a new definition of dimensionality dW=(s-2) -1 . Being related to the topological Wiener index, dW is therefore called Wiener dimensionality. Successful applications of this method to various infinite lattices (like graphene, nanocones, Sierpinski fractal triangle and carpet) testify the validity of the conjecture for infinite lattices.

We derived explicit formulae for the eccentric connectivity index and Wiener index of 2-dimensional square-octagonal TUC4C8(R) lattices with open and closed ends. New compression factors for both indices are also computed in the limit N-->∞.

A novel algorithm for the fast detection of hubs in chemical networks is presented. The algorithm identifies a set of nodes in the network as most significant, aimed to be the most effective points of distribution for fast, widespread coverage throughout the system. We show that our hubs have in general greater closeness centrality and betweenness centrality than vertices with maximal degree, while having comparable or higher degree than vertices with greatest closeness centrality and betweenness centrality. As such, they serve as all-purpose network hubs. Several theoretical and real world chemical and biological networks are tested and results are analyzed.