Title: Periodic Vertices in Graphs
|Affiliation:||University of Waterloo|
If $X$ is a graph with adjacency matrix $A$, then any question about the continuous quantum walk on $X$ is a question about the entries of the unitary matrices \[U(t) = \exp(itA)\]
Two pertinent questions are:
(a) is there a time $t$ such that some column of $U(t)$ is a scalar multiple of a standard basis vector?
(b) is there a time $t$ such that all entries in some column of $U(t)$ have the same absolute value?
(The relevant physical terms are perfect state transfer and uniform mixing respectively.) I will prove that if the answer is yes in (a), or if $X$ is bipartite and the answer is yes in (b), then the "relevant'’ eigenvalues of $X$ are integers or lie in a quadratic extension of the rationals. This can be used to prove that perfect state transfer and uniform mixing are rare phenomena.
No physics will appear in the course of the arguments, but there will be some non-trivial number theory.