Mixing of quantum walk on circulant bunkbeds

We give new observations on the mixing dynamics of a continuous-time quantum walk on circulants and their bunkbed extensions. These bunkbeds are defined through two standard graph operators: the join G + H and the Cartesian product G ⊕ H of graphs G and H . Our results include the following: • The quantum walk is average uniform mixing on circulants with bounded eigenvalue multiplicity. This extends a known fact about the cycles Cn. • Explicit analysis of the probability distribution of the quantum walk on the join of circulants. This explains why complete partite graphs are not average uniform mixing, using the fact Kn = K1 +Kn−1 and Kn,...,n = Kn + . . .+Kn. • The quantum walk on the Cartesian product of am-vertex path Pm and a circulant G, namely, Pm ⊕G, is average uniform mixing if G is. This highlights a difference between circulants and the hypercubes Qn = P2 ⊕Qn−1. Our proofs employ purely elementary arguments based on the spectra of the graphs.