Let G= V E be a simple connected graph. The distance between two vertices of G is defined to be the length of the shortest path between the two vertices. There are topological indices assigned to G and based on the distance function which are invariant under the action of the automorphism group of G. Some important indices assigned to G are the Wiener, Szeged, and PI index which we will find th...

let g be a molecular graph. the wiener index of g is defined as the summation of alldistances between vertices of g. in this paper, an exact formula for the wiener index of a newtype of nanostar dendrimer is given.

Dendrimers are highly branched organic macromolecules with successive layers of branch units surrounding a central core. The M-polynomial of nanotubes has been vastly investigated as it produces many degree-based topological indices. These indices are invariants of the topology of graphs associated with molecular structure of nanomaterials to correlate certain physicochemical properties like bo...

Precise visualization of tumor margins with characterization of microscopic tumor invasion are unmet needs in prostate oncology that demand approaches with high sensitivity and specificity. To address those needs we report surface-enhanced Raman scattering (SERS) based optical imaging for prostate cancer using a combination of live cell Raman microscopy, optimally engineered SERS tags and a ure...

general formulas are obtained for the vertex padmakar-ivan index (piv) of tetrathiafulvalene(ttf) dendrimer, whereby ttf units we are employed as branching centers. the piv index isa wiener-szeged-like index developed very recently. this topological index is defined as thesummation of all sums of nu(e) and nv(e), over all edges of connected graph g.

To overcome the current limit of tip-enhanced spectroscopy that is based on metallic nano-probes, we developed a new scanning probe with a metallic nanostar, a nanoparticle with sharp spikes. A Au nanoparticle of 5 nm was first attached to the end of a tip through DNA-DNA hybridization and mechanical pick-up. The nanoparticle was converted to a nanostar with a core diameter of ∼70 nm and spike ...

The shift of the localized surface plasmon resonance (LSPR) spectrum is widely used in bio- and chemical sensing. Traditionally, the shift is monitored at the peak maximum of the extinction spectrum. We demonstrate that the inflection point at the long wavelength side of the peak maximum shows better refractive index sensitivity than the peak maximum. A consistent improvement in bulk refractive...