Hardness results for computing optimal locally Gabriel graphs

Delaunay and Gabriel graphs are widely studied geo-metric proximity structures. Motivated by applications in wireless routing, relaxed versions of these graphs known as Locally Delaunay Graphs (LDGs) and Lo-cally Gabriel Graphs (LGGs) have been proposed. We propose another generalization of LGGs called Gener-alized Locally Gabriel Graphs (GLGGs) in the context when certain edges are forbidden in the graph. Unlike a Gabriel Graph, there is no unique LGG or GLGG for a given point set because no edge is necessarily in-cluded or excluded. This property allows us to choose an LGG/GLGG that optimizes a parameter of interest in the graph. We show that computing an edge max-imum GLGG for a given problem instance is NP-hard and also APX-hard. We also show that computing an LGG on a given point set with dilation ≤k is NP-hard. Finally, we give an algorithm to verify whether a given geometric graph G= (V, E) is a valid LGG.

Assessing the critical concentration of NH2 terminal groups on the surface of MWNTs towards chain scission of PC in PC/SAN blends: effect on dispersion, electrical conductivity and EMI shielding

The localization and dispersion quality of as received NH2 terminated multiwall carbon nanotubes (MWNT-I) and ethylene diamine (EDA) functionalized MWNTs in melt mixed blends of polycarbonate ( PC) and poly(styrene-co-acrylonitrile) (SAN) were assessed in this study using rheo-electrical and electromagnetic interference (EMI) shielding measurements. In order to improve the dispersion quality and also to selectively localize MWNTs in the PC phase of the blends, EDA was grafted onto MWNTs by two different strategies like diazonium reaction of the para-substituted benzene ring of MWNTs with EDA ( referred to as MWNT-II) and acylation of carboxyl functionalized MWNTs with thionyl chloride ( referred to as MWNT-III). By this approach we could systematically vary the concentration of NH2 functional groups on the surface of MWNTs at a fixed concentration (1 wt%) in PC/SAN blends. XPS was carried to evaluate the % concentration of N in different MWNTs and was observed to be highest for MWNT-III manifesting in a large surface coverage of EDA on the surface of MWNTs. Viscoelastic properties and melt electrical conductivities were measured to assess the dispersion quality of MWNTs using a rheo-electrical set-up both in the quiescent as well as under steady shear conditions. Rheological properties revealed chain scission of PC in the presence of MWNT-III which is due to specific interactions between EDA and PC leading to smaller PC grafts on the surface of MWNTs. The observed viscoelastic properties in the blends were further correlated with the phase morphologies under quiescent and annealed conditions. Electromagnetic interference (EMI) shielding effectiveness in X and K-u-band frequencies were measured to explore these composites for EMI shielding applications. Interestingly, MWNT-II showed the highest electrical conductivity and EMI shielding in the blends.

Electromagnetic interference shielding through MWNT grafted Fe3O4 nanoparticles in PC/SAN blends

In this study, multiwall carbon nanotubes (MWNTs) were chemically grafted onto dopamine anchored iron oxide (Fe3O4) nanoparticles via diazotization reaction to design electromagnetic (EM) shielding materials based on PC (polycarbonate)/SAN poly (styrene-co-acrylonitrile)] blends. A two step mixing protocol was adopted to selectively localize the nanoparticles in a given phase of the blends. In the first step, MWNT-g-Fe3O4 nanoparticles were solution blended with PC, followed by dilution with SAN during melt mixing in the subsequent step. This strategy, besides improving the quality of dispersion of MWNTs in the blends, facilitated enhanced EM interference shielding effectiveness (SE). Both, the MWNTs and the modified MWNTs, selectively localized in the PC phase and led to high electrical conductivity, in striking contrast to PC filled MWNT composites. The SE was measured on toroidal samples over a broad range of frequencies; X-band (8.2-12 GHz) and K-u-band (12-18 GHz). It was observed that the shielding mechanism mostly involved reflection in the blends with MWNTs, whereas absorption dominated in the case of blends with MWNT-g-Fe3O4. To realize the efficacy of this strategy, a few compositions were prepared by physical mixing MWNTs with Fe3O4 nanoparticles. Intriguingly, blends with MWNT-g-Fe3O4 nanoparticles manifested enhanced microwave absorption over physically mixed nanoparticles. An SE of -32.5 dB was observed (at 18 GHz) for MWNT (3 wt%)-g-Fe3O4 (3 vol%) in PC/SAN blends.

Microwave absorbers designed from PVDF/SAN blends containing multiwall carbon nanotubes anchored cobalt ferrite via a pyrene derivative

Lightweight and flexible electromagnetic shielding materials were designed by selectively localizing multiwall carbon nanotubes (MWNTs) anchored magnetic nanoparticles in melt mixed co-continuous blends of polyvinylidene fluoride (PVDF) and poly(styrene-co-acrylonitrile) (SAN). In order to facilitate better dispersion, the MWNTs were modified using pyrenebutyric acid (PBA) via pi-pi stacking. While one of the two-targeted properties, i.e., high electrical conductivity, was achieved by PBA modified MWNTs, high magnetic loss was accomplished by introducing nickel (NF) or cobalt ferrites (CF). Moreover, the attenuation by absorption can be tuned either by using NF (58% absorption) or CF (64% absorption) in combination with PBA-MWNTs. More interestingly, when CF was anchored on to MWNTs via the pyrene derivative, the minimum reflection loss attained was -55 dB in the Ku band (12-18 GHz) frequency and with a large bandwidth. In addition, the EM waves were blocked mostly by absorption (70%). This study opens new avenues in designing flexible and lightweight microwave absorbers.

On Locally Gabriel Geometric Graphs

Let be a set of points in the plane. A geometric graph on is said to be locally Gabriel if for every edge in , the Euclidean disk with the segment joining and as diameter does not contain any points of that are neighbors of or in . A locally Gabriel graph(LGG) is a generalization of Gabriel graph and is motivated by applications in wireless networks. Unlike a Gabriel graph, there is no unique LGG on a given point set since no edge in a LGG is necessarily included or excluded. Thus the edge set of the graph can be customized to optimize certain network parameters depending on the application. The unit distance graph(UDG), introduced by Erdos, is also a LGG. In this paper, we show the following combinatorial bounds on edge complexity and independent sets of LGG: (i) For any , there exists LGG with edges. This improves upon the previous best bound of . (ii) For various subclasses of convex point sets, we show tight linear bounds on the maximum edge complexity of LGG. (iii) For any LGG on any point set, there exists an independent set of size .