We consider throughout the text that G is a simple undirected connected graph with vertex and edge sets, V( G) and E( G), respectively. One important property of the purposed index PI v is that it simple in calculation and has similar impact as that of the Wiener index, for detail see Ashrafi and Loghman (2006). This newly defined index was a distances related index and does not coincide with Wiener index in general and in particular, for acyclic (trees) molecules. The newly defined index was abbreviated as PI v. (1998), the authors defined a new distance related molecular topological index and named it as Padmarker-Ivan index. For a thorough survey on this topic, consult the work of Mansour and Schork (2009). There are many applications of these indices which are found in chemistry, pharmaceutics, and biology. Formally, a topological index is a numeric quantity from the structural graph of a molecule. (2012).īy introducing these topological indices, the graph theory has provided the chemist with a variety of very useful tool to investigate the chemical properties of certain chemical networks. (2011), Gutman and Das (2004), Gutman and Trinajstc (1972), and Kartica et al. (2016), Ghorbani and Hosseinzadeh (2010), Graovoc et al. (2017), Ashrafi and Loghman (2008), Baig et al. The details about the work done in this direction can be found in Ali et al. Topological properties of molecular graphs in this regard are explored. There are many degree based and distance based topological indices that are defined by the mathematician and a lots of work has been done in this regard. But, the Padmaker-Ivan (PI) index is kind of the only distance related index linked to parallelism of edges. The Wiener index, due to its many applications is considered to be one of very important distance based index. Wiener applied this index to determine the physical properties of certain types of alkanes known as paraffins. It was in 1947, when Harold Wiener introduced a topological descriptor, known as the Wiener index that later become one of most useful and popular molecular descriptor. The first molecular topological index that used in chemistry was Wiener index. In this paper, the vertex PI v index of certain triangular tessellation are computed by using graph-theoretic analysis, combinatorial computing, and edge-dividing technology. Recently, the vertex Padmarkar-Ivan (PI v) index of a chemical graph G was introduced as the sum over all edges uv of a molecular graph G of the vertices of the graph that are not equidistant to the vertices u and v. They proved that the proposed PI index correlates highly with the physicochemical properties and biological activities of a large number of diverse and complex chemical compounds and the Wiener and Szeged indices. (2001), they have probed the chemical applications of the PI index. The index was firstly investigated by Khadikar et al. The PI index like other distance related indices has great disseminating power. But the Padmaker-Ivan (PI) index is kind of the only distance related index linked to parallelism of edges.
#TRIANGULAR TESSELLATION LICENSE#
This article is published under the terms of the Creative Commons Attribution License 4.The Wiener index, due to its many applications is considered to be one of very important distance-based index. International Journal of Mechanical Engineering, 5, 25-29Ĭopyright © 2020 Author(s) retain the copyright of this article. (2020) Study on the Polyhedral Triangular Tessellation of the Sphere. Regular polihedra, sphere, tessellation, subdivision methods, wrapping up patterns, library.įechete Flavia.
#TRIANGULAR TESSELLATION PATCH#
It also contains patterns to the spherical wrapping up of the polyhedral patch surface. The library contains 2 to 16-frequency tessellated icosahedral, octahedral and tetrahedral triangles. The methods have as base the subdivision of the plane polyhedral triangles or the subdivision of the projection of these triangles on the circumscribed sphere. For these methods the authors worked out a library of tessellated faces of regular polyhedra with triangular faces. There are other tessellation methods, which are difficult to compute, but enable a better approximation of the sphere, or a higher aesthetic value. This method is easy to be compute and that is why computer programmes have been develop from it.
The frequently used tessellation method consists in the subdivision of the face of the regular polyhedron and the projection of the nodes of the resulting net on the sphere.
In the last years the polyhedral triangular tessellation of the sphere was start from a regular polyhedron, especially from one with triangular faces.
Study on the Polyhedral Triangular Tessellation of the Sphere
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