Topological properties of graphs include, mainly, the number of vertices, or nodes, in a graph V, and the number of faces, or polygons, in a graph F, and the number of edges, or bonds, in a graph E...These are specified by the most important equations in the area of topology due to Euler...i.e. (a) V - E + F = 2 for the polyhedra...and (b) V - E + F = 1 for the graphs. Where V, E and F are topology characteristics of polyhedra and graphs...as opposed to geometry characteristics of polyhedra and graphs, that are not invariant under distortions of the polyhedra or graphs. Further, by reasoning that 2 faces are shared by each edge, and 2 vertices are connected together by each edge...i.e. then (c) nF = 2E & (d) pV = 2E thus, and you can transform the Euler equations (a) and (b) in to the corresponding equations due to Schlaefli which have been described elsewhere. The Schlaefli equations for the polyhedra and graphs lead naturally to a 2-dimensional Cartesian mapping construction, which has been described previously, and which leads one to a way of classifying and mapping all graphs and polyhedra and 3D networks (via the Wells point symbol construction). Further topological properties of graphs include E(2-3) & E(2-4) & E(3-3) & E(3-4) & E(3-5) etc. Here E(i-j) is the number of edges in the graph between the i-connected and j-connected vertices etc. The summation of all such i-j connected edges is just equal to E from equation (b) above. Similiarly the summation of all i- and j-connected etc. vertices in a graph is just equal to V in equation (b) above. And in the same way the faces in a graph can be divided in to a summation of 3-gons and 4-gons etc. These indexes, that further subdivide the quantities V & E & F in equation (b), in to component quantities of a given graph, are themselves topological characteristics of all graphs that are not changed by any geometrical distortions of a given graph. The primary descriptors that can be used in a structure-property analysis of graphs are thus "n", the averaged polygonality within a graph or polyhedron, and "p", the averaged connectivity within a graph or polyhedron. Together with "n: and "p", which are defined in equations (c) & (d) above, and thus by adding in the chemistry parameter "r", which is the averaged atomic number in the graph, one can map all molecules (as graphs) in the n-p-r 3D Cartesian space. From the n-p-r coordinates one can formulate various topological descriptors, like the ratios n/p and p/r etc., and the various products like np and pr and npr etc., all of which can be employed in various ways in a structure-property analysis of various groupings of chemical graphs. Further on, the subdivisions of V & E & F, in to their component parts, as has been described above in terms of "i" and "j" etc., can thus be used as a mathematically rigorous, further set of topological descriptors in a structure-property analysis. Included in such a set of topological descriptors would be the various product and ratio forms that such parameters could be combined in to thus. These descriptors or indexes of the various graphs would be mathematically rigorous as they follow directly from the equations of Euler, i.e. equations (a) & (b), and later on the corresponding equations of Schlaefli, i.e. equations (c) & (d). And as a consequence of this, such descriptors or indexes of a graph can be considered as the "natural" topological descriptors and indexes of all the varioius graphs and polyhedra thus...MJB
