  Analytic technique based on Chebyshev polynomials is developed for studying
two-dimensional lattice ribbons with hopping anisotropy. In particular, the
tight-binding models on square and triangle lattice ribbons are investigated
with anisotropic nearest neighbouring hoppings. For special values of hopping
parameters the square lattice becomes topologically equivalent to a honeycomb
one either with zigzag or armchair edges. In those cases as well as for
triangle lattices we perform the exact analytic diagonalization of
tight-binding Hamiltonians in terms of Chebyshev polynomials. Deep inside the
edge state subband the wave functions exhibit exponential spatial damping which
turns into power-law damping at edge-bulk transition point. It is shown that
strong hopping anisotropy crashes down edge states, and the corresponding
critical conditions are found.
