Topological Indexes of Carbon Allotropes and Glitter

Euler s work on the convex polyhedra resulted in an equation marking the origin of the discipline of topology, shown in one form as Eq. (1) [ 17]. 

Each edge in a convex polyhedron is shared by two faces, therefore, nF is the same as 2E. Each edge has two vertices, therefore pV is the same as 2E. By substitution, Euler s equation now reads  

Rearrangement of Eq. (2) into Eq. (3) shows that the values n, p and E must be positive integers for the expression to have validity for polyhedra 

Further restrictions are imposed on the values of n and p in order to determine the solutions to Eq. (3). In order for the number of edges, E, to be positive the values of n and p must be less than 6, and in fact n and p must be greater than two because of the impossibility of faces of zero area or spikes in the convex polyhedron. 

Substituting each of the nine combinations of 3, 4 and 5 into Eq. (3), there are only five finite, rational solutions. The five are well known and are shown as the tetrahedron, t, the octahedron, o, the icosahedron, i, the cube, c, and the dodecahedron, d, in the Table 1. 

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