The continuous-time quantum walk on the underlying graphs of association schemes have been studied, via the algebraic combinatorics structures of association schemes, namely semi-simple modules of their Bose-Mesner and (reference state dependent) Terwilliger algebras. By choosing the (walk)starting site as a reference state, the Terwilliger algebra connected with this choice turns the graph into the metric space with a distance function, hence stratifies the graph into a (d+1) disjoint union of strata (associate classes), where the amplitudes of observing the continuous-time quantum walk on all sites belonging to a given stratum are the same. Using the similarity of all vertices of underlying graph of an association scheme, it is shown that the transition probabilities between the vertices depend only on the distance between the vertices (kind of relations or association classes). Hence for a continuous-time quantum walk over a graph associated with a given scheme with diameter d, we have exactly (d + 1) different transition probabilities (i.e., the number of strata or number of distinct eigenvalues of adjacency matrix).In graphs of association schemes with known spectrum, namely with relevant BoseMesner algebras of known eigenvalues and eigenstates, the transition amplitudes and average probabilities are given in terms of dual eigenvalues of association schemes. As most of association schemes arise from finite groups, hence the continuous-time walk

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.

In papers [1,2], the amplitudes of continuous-time quantum walk on graphs possessing quantum decomposition (QD graphs) have been calculated by a new method based on spectral distribution associated to their adjacency matrix. Here in this paper, it is shown that the continuous-time quantum walk on any arbitrary graph can be investigated by spectral distribution method, simply by using Krylov subspace-Lanczos algorithm to generate orthonormal bases of Hilbert space of quantum walk isomorphic to orthogonal polynomials. Also new type of graphs possessing generalized quantum decomposition have been introduced, where this is achieved simply by relaxing some of the constrains imposed on QD graphs and it is shown that both in QD and GQD graphs, the unit vectors of strata are identical with the orthonormal basis produced by Lanczos algorithm.Moreover, it is shown that probability amplitude of observing walk at a given vertex is proportional to its coefficient in the corresponding unit vector of its stratum, and it can be written in terms of the amplitude of its stratum. Finally the capability of Lanczos-based algorithm for evaluation of walk on arbitrary graphs ( GQD or non-QD types), has been tested by calculating the probability amplitudes of quantum walk on some interesting finite (infinite) graph of GQD type and finite (infinite) path graph of non-GQD type, where the asymptotic behavior of the probability amplitudes at infinite limit of number of vertices, are in agreement with those of central limit theorem of Ref. [3].