To use the Zagreb Group and Aihara definition of resonance
energy, it is necessary that the roots of the reference polynomial
all be real. A partial proof that they are has been obtained in three
ways. Direct solution of the reference polynomial for annulenes
shows all roots real in this case. Application of Sturm sequences
promises the complete proof in principle, but requires the proof of
inequalities which we have so far resolved only for molecules with
four or fewer atoms. A graph theoretical approach succeeds for all
conjugated hydrocarbons in which no edge is shared by two rings.
It is also suggested that the reference polynomial may be used for
discriminating planar isospectral molecules
