On the remoteness function in median graphs

A profile on a graph G is any nonempty multiset whose elements are vertices from G. The corresponding remoteness function associates to each vertex x 2 V.G/ the sum of distances from x to the vertices in the profile. Starting from some nice and useful properties of the remoteness function in hypercubes, the remoteness function is studied in arbitrary median graphs with respect to their isometric embeddings in hypercubes. In particular, a relation between the vertices in a median graph G whose remoteness function is maximum (antimedian set of G) with the antimedian set of the host hypercube is found. While for odd profiles the antimedian set is an independent set that lies in the strict boundary of a median graph, there exist median graphs in which special even profiles yield a constant remoteness function. We characterize such median graphs in two ways: as the graphs whose periphery transversal number is 2, and as the graphs with the geodetic number equal to 2. Finally, we present an algorithm that, given a graph G on n vertices and m edges, decides in O.mlog n/ time whether G is a median graph with geodetic number 2

Development of Elastomeric Hybrid Composite Based on Synthesised Manosilica and Short Nylon Fiber

Nanoscale silica was synthesized by precipitation method using sodium silicate and dilute hydrochloric acid under controlled conditions. The synthesized silica was characterized by Scanning Electron Microscopy (SEM), Transmission Electron Microscopy (TEM), BET adsorption and X-Ray Diffraction (XRD). The particle size of silica was calculated to be 13 nm from the XRD results and the surface area was found to be 295 m2/g by BET method. The performance of this synthesized nanosilica as a reinforcing filler in natural rubber (NR) compound was investigated. The commercial silica was used as the reference material. Nanosilica was found to be effective reinforcing filler in natural rubber compound. Filler-matrix interaction was better for nanosilica than the commercial silica. The synthesized nanosilica was used in place of conventional silica in HRH (hexamethylene tetramine, resorcinol and silica) bonding system for natural rubber and styrene butadiene rubber / Nylon 6 short fiber composites. The efficiency of HRH bonding system based on nanosilica was better. Nanosilica was also used as reinforcing filler in rubber / Nylon 6 short fiber hybrid composite. The cure, mechanical, ageing, thermal and dynamic mechanical properties of nanosilica / Nylon 6 short fiber / elastomeric hybrid composites were studied in detail. The matrices used were natural rubber (NR), nitrile rubber (NBR), styrene butadiene rubber (SBR) and chloroprene rubber (CR). Fiber loading was varied from 0 to 30 parts per hundred rubber (phr) and silica loading was varied from 0 to 9 phr. Hexa:Resorcinol:Silica (HRH) ratio was maintained as 2:2:1. HRH loading was adjusted to 16% of the fiber loading. Minimum torque, maximum torque and cure time increased with silica loading. Cure rate increased with fiber loading and decreased with silica content. The hybrid composites showed improved mechanical properties in the presence of nanosilica. Tensile strength showed a dip at 10 phr fiber loading in the case of NR and CR while it continuously increased with fiber loading in the case of NBR and SBR. The nanosilica improved the tensile strength, modulus and tear strength better than the conventional silica. Abrasion resistance and hardness were also better for the nanosilica composites. Resilience and compression set were adversely affected. Hybrid composites showed anisotropy in mechanical properties. Retention in ageing improved with fiber loading and was better for nanosilica-filled hybrid composites. The nanosilica also improved the thermal stability of the hybrid composite better than the commercial silica. All the composites underwent two-step thermal degradation. Kinetic studies showed that the degradation of all the elastomeric composites followed a first-order reaction. Dynamic mechanical analysis revealed that storage modulus (E’) and loss modulus (E”) increased with nanosiica content, fiber loading and frequency for all the composites, independent of the matrix. The highest rate of increase was registered for NBR rubber.