Many physico-chemical parameters of molecules are determined by or are dependent upon their HOMO–LUMO gaps, as has become of special interest for conjugated-carbon nano-structures obtained from graphene and its congeners. Here, we deduce an elegant yet simple upper-bound estimate to the HOMO–LUMO gaps for such molecular π -networks corresponding to connected subgraphs of graphene. The result elucidates the general situations (involving larger fragments) for which the HOMO–LUMO gap is small.