Calculating Topological Indexes of Networks from the Corresponding Wells Point Symbol
Michael J. Bucknum · SSRN Electronic Journal · 2002
This paper begins with a review of the Euler relation for the convex polyhedra. The Schlafli relation is derived from this by introducing the secondary topological indexes of polygonality and connectivity. A topology map of the polyhedra and extended structures, in a Schlafli space, is illustrated and is discussed from the point of view of its organizational value for defining the topological relationship of structures from their Schlafli indexes. A comment is then made with respect to the definition of the various vertex connectivities in structures, from knowledge of the circuit number of the vertex. It is shown here that vertices containing polar connections have a reduced circuit number from that calculated according to the prescription p(p-1)/2. Next a review is made of the Wells point symbols and corresponding Schlafli indexes; first of the regular diamond and graphene nets; second of the semi-regular nets, including the Archimedean fullerene polyhedron and the Cooperite structure-type, and the Catalan fluorite and Waserite structure-types; and third of the irregular nets to include the Wellsean glitter and phenacite structure-types. The direct translation of the Wells point symbol to a weighted average polygonality, n, and a weighted average connectivity, p, is demonstrated in these examples.