  The continuous-time quantum walk (CTQW) on root lattice $A_n$ (known as
hexagonal lattice for $n=2$) and honeycomb one is investigated by using
spectral distribution method. To this aim, some association schemes are
constructed from abelian group $Z^{\otimes n}_m$ and two copies of finite
hexagonal lattices, such that their underlying graphs tend to root lattice
$A_n$ and honeycomb one, as the size of the underlying graphs grows to
infinity. The CTQW on these underlying graphs is investigated by using the
spectral distribution method and stratification of the graphs based on
Terwilliger algebra, where we get the required results for root lattice $A_n$
and honeycomb one, from large enough underlying graphs. Moreover, by using the
stationary phase method, the long time behavior of CTQW on infinite graphs is
approximated with finite ones. Also it is shown that the Bose-Mesner algebras
of our constructed association schemes (called $n$-variable $P$-polynomial) can
be generated by $n$ commuting generators, where raising, flat and lowering
operators (as elements of Terwilliger algebra) are associated with each
generator. A system of $n$-variable orthogonal polynomials which are special
cases of \textit{generalized} Gegenbauer polynomials is constructed, where the
probability amplitudes are given by integrals over these polynomials or their
linear combinations. Finally the suppersymmetric structure of finite honeycomb
lattices is revealed. Keywords: underlying graphs of association schemes,
continuous-time quantum walk, orthogonal polynomials, spectral distribution.
PACs Index: 03.65.Ud
