The present paper investigates arc-transtive graphs in term of their stability, and shows, somewhat contrary to expectations, that the property of instability is not as rare as previously thought. Until quite recently, the only known example of a finite, arc-transitive vertex-determining unstable graph was the underlying graph of the dodecahedron. This paper illustrates some methods for constru...

We provide a process to extend any bipartite diametrical graph of diameter 4 to an S-graph of the same diameter and partite sets. For a bipartite diametrical graph of diameter 4 and partite sets U and W , where 2m |U| ≤ |W |, we prove that 2 is a sharp upper bound of |W | and construct an S-graph G 2m, 2 in which this upper bound is attained, this graph can be viewed as a generalization of the ...

Coverage and connectivity issues of three-dimensional (3D) networks are addressed in [2], but that work assumes that a node can be placed at any arbitrary location. In this work, we drop that assumption and rather assume that nodes are uniformly and densely deployed in a 3D space. We want to devise a mechanism that keeps some nodes active and puts other nodes into sleep so that the number of ac...

Polyoxometalates (POMs) have been a popular motif in crystal engineering. In this work, seven ionic crystals based on Keggin- or Dawson-type POMs and mononuclear Bi(III) complexes were synthesized. The center exhibited triangular dodecahedron, square antiprism, pseudo-cubic eight-coordination geometries, with DMSO DMF as ligands. showed polymorphic transformations, high proton conductivities de...

The ground-state electronic and structural properties, and the electronic excitations of the lowest energy isomers of the Ag8 cluster are calculated using density functional theory (DFT) and timedependent DFT (TDDFT) in real time and real space scheme, respectively. The optical spectra provided by TDDFT predict that the D2d dodecahedron isomer is the structural minimum of Ag8 cluster. Indeed, i...

Simple equations are derived that give the shapes of various convex polyhedra. The five regular polyhedra, called Platonic solids (the tetrahedron, hexahedron or cube, octahedron, dodecahedron and icosahedron), and polyhedra composed of crystallographically low-index planes are treated. The equations also give shapes that are nearly polyhedral with round edges, or intermediate shapes between a ...

We present the necessary and sufficient conditions for stability and instability of the stationary rotation of a system of n identical point vortices located at the same latitude on a sphere at vertices of a regular n-gon. We also examine stability of the equilibrium configuration of identical point vortices, situated at the vertices of a regular polyhedra. It is proved that vortex tetrahedron,...

BACKGROUND Despite being the conventional choice, whole body PET cameras with a 76 cm diameter ring are not the optimal means of human brain imaging. OBJECTIVE In fact, a dedicated brain PET with a better geometrical structure has the potential to achieve a higher sensitivity, a higher signal-to-noise ratio, and a better imaging performance. METHODS In this study, a polyhedron geometrical d...

We have developed a new green chemical approach for the shape-controlled synthesis of single-crystalline hematite nanocrystals in aqueous medium. FESEM, HRTEM and SAED techniques were used to determine the morphology and crystallographic orientations of each nanocrystal and its exposed facets. PXRD and HRTEM techniques revealed that the nanocrystals are single crystalline in nature; twins and s...