Algebraic graph theory, eigenspaces, quantum walks
The rank of the average mixing matrix of trees with all eigenvalues distinct, is investigated. The rank of the average mixing matrix of a tree on n vertices with n distinct eigenvalues is bounded above by ⌈n/2⌉. Computations on trees up to 20 vertices suggest that the rank attains this upper bound most of the time. An infinite family of trees whose average mixing matrices have ranks which are bounded away from this upper bound, is given. A lower bound on the rank of the average mixing matrix of a tree, is also given.
Godsil, Chris; Guo, Krystal; and Sinkovic, John.
"Average Mixing Matrix of Trees",
Electronic Journal of Linear Algebra,
Volume 34, pp. 269-282.