105 research outputs found
Universal State Transfer on Graphs
A continuous-time quantum walk on a graph is given by the unitary matrix
, where is the Hermitian adjacency matrix of . We say
has pretty good state transfer between vertices and if for any
, there is a time , where the -entry of
satisfies . This notion was introduced by Godsil
(2011). The state transfer is perfect if the above holds for . In
this work, we study a natural extension of this notion called universal state
transfer. Here, state transfer exists between every pair of vertices of the
graph. We prove the following results about graphs with this stronger property:
(1) Graphs with universal state transfer have distinct eigenvalues and flat
eigenbasis (where each eigenvector has entries which are equal in magnitude).
(2) The switching automorphism group of a graph with universal state transfer
is abelian and its order divides the size of the graph. Moreover, if the state
transfer is perfect, then the switching automorphism group is cyclic. (3) There
is a family of prime-length cycles with complex weights which has universal
pretty good state transfer. This provides a concrete example of an infinite
family of graphs with the universal property. (4) There exists a class of
graphs with real symmetric adjacency matrices which has universal pretty good
state transfer. In contrast, Kay (2011) proved that no graph with real-valued
adjacency matrix can have universal perfect state transfer. We also provide a
spectral characterization of universal perfect state transfer graphs that are
switching equivalent to circulants.Comment: 27 pages, 3 figure
Fractional revival on Cayley graphs over abelian groups
In this paper, we investigate the existence of fractional revival on Cayley
graphs over finite abelian groups. We give a necessary and sufficient condition
for Cayley graphs over finite abelian groups to have fractional revival. As
applications, the existence of fractional revival on circulant graphs and
cubelike graphs are characterized
- …