Virtually Abelian Quantum Walks

We introduce quantum walks on Cayley graphs of non-Abelian groups. We focus on the easiest case of virtually Abelian groups, and introduce a technique to reduce the quantum walk to an equivalent one on an Abelian group with coin system having larger dimension. We apply the technique in the case of two quantum walks on virtually Abelian groups with planar Cayley graphs, finding the exact solution.Comment: 10 pages, 3 figure

Finite 33-connected homogeneous graphs

A finite graph \G is said to be {\em $(G,3)$-$($connected$)$ homogeneous} if every isomorphism between any two isomorphic (connected) subgraphs of order at most $3$ extends to an automorphism $g\in G$ of the graph, where $G$ is a group of automorphisms of the graph. In 1985, Cameron and Macpherson determined all finite $(G, 3)$-homogeneous graphs. In this paper, we develop a method for characterising $(G,3)$-connected homogeneous graphs. It is shown that for a finite $(G,3)$-connected homogeneous graph \G=(V, E), either G_v^{\G(v)} is $2$--transitive or G_v^{\G(v)} is of rank $3$ and \G has girth $3$, and that the class of finite $(G,3)$-connected homogeneous graphs is closed under taking normal quotients. This leads us to study graphs where $G$ is quasiprimitive on $V$. We determine the possible quasiprimitive types for $G$ in this case and give new constructions of examples for some possible types

Automorphism groups of tetravalent Cayley graphs on minimal non-abelian groups

A Cayley graph X = Cay(G, S) is called normal for G if the right regular representation R(G) of G is normal in the full automorphism group Aut(X) of X. In the present paper it is proved that all connected tetravalent Cayley graphs on a minimal non-abelian group G are normal when (|G|,2) = (|G|,3) = 1, and X is not isomorphic to either Cay(G, S), where |G| = 5n, and |Aut(X)| = 2m.3.5n, where m ∈ {2,3} and n ≥ 3, or Cay(G, S) where |G| = 5qn (q is prime) and |Aut(X)| = 2m.3.5.qn, where q ≥ 7, m ∈ {2,3} and n ≥ 1