Let G be a simple graph on n vertices with the eigenvalues (of an adjacency matrix) λ₁ ≥ λ₂ ≥ ... ≥ λ_n. For 1 ≤ i < j ≤ n, the (i,j)-spectral spread (or just (i,j)-spread) of G is defined as the difference λ_i - λ_j. The program Graffiti, developed by S. Fajtlowicz, posed the conjecture that the (1,2)-spread of fullerenes is at most 1. Here we prove this conjecture by using the interlacing theorem in an interesting manner and then extend this method to show that the dodecahedron has the largest (1,2)-spread amongst all fullerenes.