Enumeration of networks

Abstract

AbstractThe results of previous workers relating the number of internal and external vertices in a two‐dimensional network to the number of elements of one given geometry are generalized to cover elements of any geometry (not necessarily convex) and any number of edge nodes, which may be vertex nodses of adjacent elements.It is shown that a relationship between the number of elements and number of vertices (nodes) alone exists only on a surface (plane or otherwise) and therefore cannot be extended to three‐dimensional networks.In three‐dimensions, the number of elements with any number of edges and vertices is related to the number of internal and external edges and vertices, is related to the number of internal and external edges and vertices, enclosed by a surface which may be non‐simple, (i.e. of topological genus greater than zero).