  In quasi-1D $\pi$-conjugated polymers such as \emph{trans}-polyacetylene and
polyenes, electron correlation effects determine the "reversed" excited state
ordering in which the lowest two-photon $2A_{g}$ state lies below the lowest
one-photon $1B_{u}$ state. In this work, we present conclusive theoretical
evidence of reversed excited state ordering in fairly 2D $\pi$-conjugated
systems, namely, diamond-shaped graphene quantum dots (DQDs). Our electron
correlated calculations show that DQDs begin to exhibit reversed excited
ordering with increasing size, in disagreement with independent-particle
picture. This signals the onset of strong correlation effects which renders
them nonluminescent. Further, we calculate and analyze the two-photon
absorption (TPA) spectra as well as photoinduced absorption (PA) spectra of
these systems and find excellent agreement with the available experimental
results. Our investigations demonstrate that unlike a strictly 1D system
like\emph{trans}-polyacetylene, the non-linear and excited state absorptions in
DQDs are highly intricate, with several even parity states responsible for
strong absorptions. Our results could play an important role in the design of
graphene-based non-linear optical devices.
