  We study the single electron model of a semi-infinite graphene sheet
interfaced with the vacuum and terminated along a zigzag edge. The model is a
Schroedinger operator acting on $L^2(\mathbb{R}^2)$: $H^\lambda_{\rm
edge}=-\Delta+\lambda^2 V_\sharp$, with a potential $V_\sharp$ given by a sum
of translates an atomic potential well, $V_0$, of depth $\lambda^2$, centered
on a subset of the vertices of a discrete honeycomb structure with a zigzag
edge. We give a complete analysis of the low-lying energy spectrum of
$H^\lambda_{\rm edge}$ in the strong binding regime ($\lambda$ large). In
particular, we prove scaled resolvent convergence of $H^\lambda_{\rm edge}$
acting on $L^2(\mathbb{R}^2)$, to the (appropriately conjugated) resolvent of a
limiting discrete tight-binding Hamiltonian acting in
$l^2(\mathbb{N}_0;\mathbb{C}^2)$. We also prove the existence of {\it edge
states}: solutions of the eigenvalue problem for $H^\lambda_{\rm edge}$ which
are localized transverse to the edge and pseudo-periodic (propagating or
plane-wave like) parallel to the edge. These edge states arise from a
"flat-band" of eigenstates the tight-binding Hamiltonian.
