Discrete quantum walks with marked vertices and their average vertex mixing matrices

We study the discrete quantum walk on a regular graph X that assigns negative identity coins to marked vertices S and Grover coins to the unmarked ones. We find combinatorial bases for the eigenspaces of the transition matrix, and derive a formula for the average vertex mixing matrix $\hat{M}$.

We then find bounds for entries in $\hat{M}$, and study when these bounds are tight. In particular, the average probabilities between marked vertices are lower bounded by a matrix determined by the induced subgraph $X\left[S\right]$, the vertex-deleted subgraph $X﹨S$, and the edge deleted subgraph $X-E\left(S\right)$. We show this bound is achieved if and only if the marked vertices have walk-equitable neighborhoods in the vertex-deleted subgraph. Finally, for quantum walks attaining this bound, we determine when $\hat{M}[S,S]$ is symmetric, positive semidefinite or uniform.