Icosahedral fulleroids

In this chapter, which is an adaptation of [DeDeOO], are considered icosahedral fulleroids (or I-fulleroids, or, more precisely, 1(5, byfulleroids, i.e. ( 5, b], 3)-spheres of symmetry I or / ). For some values of b, the smallest such fulleroids are indicated and their unicity is proved. Also, several infinite series of them are presented. 

Here CV(G) stands for a ( 5,6, 3)-sphere with v vertices and symmetry group G. Although this notation is not generally unique, it will suffice for our purpose. 

All 7-fiilleroids known so far and simple ways to describe them are given in Section 19.1 based on that some infinite series are introduced. In Section 19.2, a necessary condition for the p-vectors, which implies that five of the new 7-fulleroids are minimal for their respective values of v, is derived. 

Note that for b = 12, the smallest p-vector, which fulfills the condition of Lemma 19.2.3, is not realizable. 

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DeDeOO] O. Delgado-Friedrichs and M. Deza, More icosahedral fulleroids, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 51 (2000) 97-115. 

JeTrOl] S. Jendrol and M. Trenkler, More icosahedral fulleroids, Journal of Mathematical Chemistry 29 (2001) 235-243. 

The case b = 6 is the classical fullerene case. Theorem 2.2.2 gives that all 7(5,6)-fulleroids, i.e. fullerenes of icosahedral symmetry, are of the form GCk,i(Dodecahedron). See on Figure 19.1 the first three of the following smallest icosahedral fullerenes besides Dodecahedron ... 

O.D. Friedrichs and M. Deza, More Icosahedral Symmetry Fulleroids, in Discrete Mathematical Chemistry, DIMACS Series in Discrete Mathematical and Theoretical Computer Science, Vol. 51, eds. P. Hansen, P. W. Fowler and M. Zheng, American Mathematical Society, Providence, RI, 2000, pp. 97-116. 

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