A supercharacter theory for a finite group G is a set of superclasses each of which is a union of conjugacy classes together with a set of sums of irreducible characters called supercharacters that together satisfy certain compatibility conditions. The aim of this paper is to give a description of some finite simple groups with exactly three or four supercharacter theories.

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The eccentricity sequence of G, denoted by e G , is the set of all e v i . In this paper, the eccentricity sequences of two infinite classes of fullerenes are computed. As a consequence, the eccentric connectivity indices of these fullerenes are computed.