Heilbronner modes and the line graphs

We restrict attention to graphs G that have no pendant edges. For concreteness, take a graph G corresponding to a trivalent polyhedron P, where each edge has four 

if the local pattern of coefficients is part of a Heilbronner mode, we have 

In the case of the line graph of a fiillerene C , the spectrum consists of the entire adjacency spectrum of the fullerene itself (shifted by a constant), 

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