Betweenness Centrality in Some Classes of Graphs

There are several centrality measures that have been introduced and studied for real world networks. They account for the different vertex characteristics that permit them to be ranked in order of importance in the network. Betweenness centrality is a measure of the influence of a vertex over the flow of information between every pair of vertices under the assumption that information primarily flows over the shortest path between them. In this paper we present betweenness centrality of some important classes of graphs.

Size-dependent magnetization fluctuations in NiO nanoparticles

The finite size and surface roughness effects on the magnetization of NiO nanoparticles is investigated. A large magnetic moment arises for an antiferromagnetic nanoparticle due to these effects. The magnetic moment without the surface roughness has a non-monotonic and oscillatory dependence on $R$, the size of the particles, with the amplitude of the fluctuations varying linearly with $R$. The geometry of the particle also matters a lot in the calculation of the net magnetic moment. An oblate spheroid shape particle shows an increase in net magnetic moment by increasing oblateness of the particle. However, the magnetic moment values thus calculated are very small compared to the experimental values for various sizes, indicating that the bulk antiferromagnetic structure may not hold near the surface. We incorporate the surface roughness in two different ways; an ordered surface with surface spins inside a surface roughness shell aligned due to an internal field, and a disordered surface with randomly oriented spins inside surface roughness shell. Taking a variational approach we find that the core interaction strength is modified for nontrivial values of $Delta$ which is a signature of multi-sublattice ordering for nanoparticles. The surface roughness scale $Delta $ is also showing size dependent fluctuations, with an envelope decay $Deltasim R^{-1/5}$. The net magnetic moment values calculated using spheroidal shape and ordered surface are close to the experimental values for different sizes.

Origin of high piezoelectric response of Pb(Zr_xTi_1-x)O_3 at the morphotropic phase boundary: Role of elastic instability

Temperature dependent structural changes in a nearly pure monoclinic phase composition (x=0.525) of Pb(Zr_xTi_1-x)O_3 (PZT) have been investigated using Rietveld analysis of high-resolution synchrotron powder x-ray diffraction data and correlated with changes in the dielectric constant and planar electromechanical coupling coefficient. Our results show that the intrinsic piezoelectric response of the tetragonal phase of PZT is higher than that of the monoclinic phase. It is also shown that the high piezoelectric response of PZT may be linked with an anomalous softening of the elastic modulus (1/S_11) of the tetragonal compositions closest to the morphotropic phase boundary.