Proyecciones Journal of Mathematics
Vol. 29, N° 1, pp. 31-39, May 2010.
Universidad Católica del Norte
Antofagasta - Chile


GRAPHS r-POLAR SPHERICAL REALIZATION*


Eduardo Montenegro
Eduardo Cabrera
José González
Alejandro Nettle
Ramón Robres

Universidad De Playa Ancha, Chile


Correspondencia a:

Abstract

The graph to considered will be in general simple and finite, graphs with a nonempty set of edges. For a graph G, V(G) denote the set of vertices and E(G) denote the set of edges. Now, let P_r = (0, 0, 0, r) ∈ R⁴, r ∈ R⁺ . The r-polar sphere, denoted by S_Pr , is defined by {x ∈ R⁴/ ||x|| = 1 Λ x ≠ P_r }: The primary target of this work is to present the concept of r-Polar Spherical Realization of a graph. That idea is the following one: If G is a graph and h : V (G) ? S_Pr is a injective function, them the r-Polar Spherical Realization of G, denoted by G*, it is a pair (V (G*), E(G*)) so that V (G*) = {h(v)/v ∈ V (G)} and E(G*) = {arc(h(u)h(v))/uv ∈ E(G)}, in where arc(h(u)h(v)) it is the arc of curve contained in the intersection of the plane defined by the points h(u), h(v), P_r and the r-polar sphere.

Keywords : Graph, Sphere.

AMS Subject Classifications : 05C25 ; 05C35.

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Received : October 2009. Accepted : January 2010

Eduardo Montenegro
Departamento de Matemáticas y Física
Facultad de Ciencias Naturales y Exactas
Universidad de Playa Ancha
Casilla 34 V
Valparaíso
e-mail : emontene@upla.cl


Eduardo Cabrera
Departamento de Matemáticas y Física
Facultad de Ciencias Naturales y Exactas
Universidad de Playa Ancha
Casilla 34 V
Valparaíso
e-mail : ecabrera@upla.cl


José González
Departamento de Matemáticas y Física
Facultad de Ciencias Naturales y Exactas
Universidad de Playa Ancha
Casilla 34 V
Valparaíso
e-mail : egonzalez@upla.cl


Alejandro Nettle
Departamento de Matemáticas y Física
Facultad de Ciencias Naturales y Exactas
Universidad de Playa Ancha
Casilla 34 V
Valparaíso
e-mail : anettle@upla.cl


Ramón Robres
Departamento de Matemáticas y Física
Facultad de Ciencias Naturales y Exactas
Universidad de Playa Ancha
Casilla 34 V
Valparaíso
e-mail :rrobres@upla.cl