  Using the spectral distribution associated with the adjacency matrix of
graphs, we introduce a new method of calculation of amplitudes of
continuous-time quantum walk on some rather important graphs, such as line,
cycle graph $C_n$, complete graph $K_n$, graph $G_n$, finite path and some
other finite and infinite graphs, where all are connected with orthogonal
polynomials such as Hermite, Laguerre, Tchebichef and some other orthogonal
polynomials. It is shown that using the spectral distribution, one can obtain
the infinite time asymptotic behavior of amplitudes simply by using the method
of stationary phase approximation(WKB approximation), where as an example, the
method is applied to star, two-dimensional comb lattices, infinite Hermite and
Laguerre graphs. Also by using the Gauss quadrature formula one can approximate
infinite graphs with finite ones and vice versa, in order to derive large time
asymptotic behavior by WKB method. Likewise, using this method, some new graphs
are introduced, where their amplitude are proportional to product of amplitudes
of some elementary graphs, even though the graphs themselves are not the same
as Cartesian product of their elementary graphs. Finally, via calculating mean
end to end distance of some infinite graphs at large enough times, it is shown
that continuous time quantum walk at different infinite graphs belong to
different universality classes which are also different than those of the
corresponding classical ones.
