Rainbow connection number of Cm o Pn and Cm o Cn

Let <em>G </em>= (<em>V</em>(<em>G</em>),<em>E</em>(<em>G</em>)) be a nontrivial connected graph. A rainbow path is a path which is each edge colored with different color. A rainbow coloring is a coloring which any two vertices should be joined by at least one rainbow path. For two different vertices, <em>u,v</em> in <em>G</em>, a geodesic path of <em>u-v</em> is the shortest rainbow path of <em>u-v</em>. A strong rainbow coloring is a coloring which any two vertices joined by at least one rainbow geodesic. A rainbow connection number of a graph, denoted by <em>rc</em>(<em>G</em>), is the smallest number of color required for graph <em>G</em> to be said as rainbow connected. The strong rainbow color number, denoted by <em>src</em>(<em>G</em>), is the least number of color which is needed to color every geodesic path in the graph <em>G</em> to be rainbow. In this paper, we will determine the rainbow connection and strong rainbow connection for Corona Graph <em>Cm</em> o <em>Pn</em>, and <em>Cm</em> o <em>Cn</em>.

Geometric modeling of knitted fabrics using helicoid scaffolds

We present a bicontinuous, minimal surface (the helicoid) as a scaffold on which to define the topology and geometry of yarns in a weft-knitted fabric. Modeling with helicoids offers a geometric approach to simulating a physical manufacturing process, which should generate geometric models suitable for downstream mechanical and validity analyses. The centerline of a yarn in a knitted fabric is specified as a geodesic path, with constrained boundary conditions, running along a helicoid at a fixed distance. The shape of the yarn’s centerline is produced via an optimization process over a polyline. The distances between the vertices of the polyline are shortened and a repulsive potential keeps the vertices at a specified distance from the helicoid. These actions and constraints are formulated into a single “energy” function, which is then minimized. The yarn geometry is generated as a tube around the centerline. The optimized configuration, defined for a half loop, is duplicated, reflected, and shifted to produce the centerlines for the multiple stitches that make up a fabric. In addition, the parameters of the helicoid may be used to control the size and shape of the fabric’s stitches. We show that helicoid scaffolds may be used to define both knit and purl stitches, which are then combined to produce models of all-knit, rib, and garter fabrics.

Convexity in complex networks

Metric graph properties lie in the heart of the analysis of complex networks, while in this paper we study their convexity through mathematical definition of a convex subgraph. A subgraph is convex if every geodesic path between the nodes of the subgraph lies entirely within the subgraph. According to our perception of convexity, convex network is such in which every connected subset of nodes induces a convex subgraph. We show that convexity is an inherent property of many networks that is not present in a random graph. Most convex are spatial infrastructure networks and social collaboration graphs due to their tree-like or clique-like structure, whereas the food web is the only network studied that is truly non-convex. Core–periphery networks are regionally convex as they can be divided into a non-convex core surrounded by a convex periphery. Random graphs, however, are only locally convex meaning that any connected subgraph of size smaller than the average geodesic distance between the nodes is almost certainly convex. We present different measures of network convexity and discuss its applications in the study of networks.

Geodesic structure of magnetically charged regular black hole

The purpose of this paper is to study the geodesic structure of magnetically charged regular black hole (MCRBH). The behavior of timelike and null geodesics of MCRBH is investigated. The graphs have been plotted to show the relation between distance versus time and proper time for photon-like and massive particle. For radial and circular motion, the effective potential has been plotted with different parameters of BH. We conclude that massive particles move around the BH in timelike geodesic path.