8,639 research outputs found
Nonlocal Games and Quantum Permutation Groups
We present a strong connection between quantum information and quantum
permutation groups. Specifically, we define a notion of quantum isomorphisms of
graphs based on quantum automorphisms from the theory of quantum groups, and
then show that this is equivalent to the previously defined notion of quantum
isomorphism corresponding to perfect quantum strategies to the isomorphism
game. Moreover, we show that two connected graphs and are quantum
isomorphic if and only if there exists and that are
in the same orbit of the quantum automorphism group of the disjoint union of
and . This connection links quantum groups to the more concrete notion
of nonlocal games and physically observable quantum behaviours. We exploit this
link by using ideas and results from quantum information in order to prove new
results about quantum automorphism groups, and about quantum permutation groups
more generally. In particular, we show that asymptotically almost surely all
graphs have trivial quantum automorphism group. Furthermore, we use examples of
quantum isomorphic graphs from previous work to construct an infinite family of
graphs which are quantum vertex transitive but fail to be vertex transitive,
answering a question from the quantum group literature.
Our main tool for proving these results is the introduction of orbits and
orbitals (orbits on ordered pairs) of quantum permutation groups. We show that
the orbitals of a quantum permutation group form a coherent
configuration/algebra, a notion from the field of algebraic graph theory. We
then prove that the elements of this quantum orbital algebra are exactly the
matrices that commute with the magic unitary defining the quantum group. We
furthermore show that quantum isomorphic graphs admit an isomorphism of their
quantum orbital algebras which maps the adjacency matrix of one graph to that
of the other.Comment: 39 page
Groups all of whose undirected Cayley graphs are integral
Let be a finite group, be a set such that if
, then , where denotes the identity element of .
The undirected Cayley graph of over the set is the graph
whose vertex set is and two vertices and are adjacent whenever
. The adjacency spectrum of a graph is the multiset of all
eigenvalues of the adjacency matrix of the graph. A graph is called integral
whenever all adjacency spectrum elements are integers. Following Klotz and
Sander, we call a group Cayley integral whenever all undirected Cayley
graphs over are integral. Finite abelian Cayley integral groups are
classified by Klotz and Sander as finite abelian groups of exponent dividing
or . Klotz and Sander have proposed the determination of all non-abelian
Cayley integral groups. In this paper we complete the classification of finite
Cayley integral groups by proving that finite non-abelian Cayley integral
groups are the symmetric group of degree , and
for some integer , where is the
quaternion group of order .Comment: Title is change
Commuting involution graphs for [(A)\tilde]n
In this article we consider the commuting graphs of involution conjugacy classes in the affine Weyl group A~n. We show that where the graph is connected the diameter is at most 6.
MSC(2000): 20F55, 05C25, 20D60
A note on commuting graphs for symmetric groups
The commuting graph C(G;X) , where G is a group and X a subset of G, has X as its vertex set with two distinct elements of X joined by an edge when they commute in G. Here the diameter and disc structure of C(G;X) is investigated when G is the symmetric group and X a conjugacy class of
G
- …