The separator of a graph is the difference
between the two largest eigenvalues of its adjacency matrix.
The program {\em Graffiti}, developed by S.~Fajtlowicz,
posed the conjecture that the separator of any fullerene is at most~$1$.
Here we prove this conjecture by using the {\it interlacing theorem}
in an interesting manner and then extend this method to show that the
dodecahedron has the largest separator amongst all fullerenes.
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