4,011 research outputs found
Automorphism Groups of Geometrically Represented Graphs
We describe a technique to determine the automorphism group of a
geometrically represented graph, by understanding the structure of the induced
action on all geometric representations. Using this, we characterize
automorphism groups of interval, permutation and circle graphs. We combine
techniques from group theory (products, homomorphisms, actions) with data
structures from computer science (PQ-trees, split trees, modular trees) that
encode all geometric representations.
We prove that interval graphs have the same automorphism groups as trees, and
for a given interval graph, we construct a tree with the same automorphism
group which answers a question of Hanlon [Trans. Amer. Math. Soc 272(2), 1982].
For permutation and circle graphs, we give an inductive characterization by
semidirect and wreath products. We also prove that every abstract group can be
realized by the automorphism group of a comparability graph/poset of the
dimension at most four
Complete bipartite graphs whose topological symmetry groups are polyhedral
We determine for which , the complete bipartite graph has an
embedding in whose topological symmetry group is isomorphic to one of the
polyhedral groups: , , or .Comment: 25 pages, 6 figures, latest version has minor edits in preparation
for submissio
Clifford quantum computer and the Mathieu groups
One learned from Gottesman-Knill theorem that the Clifford model of quantum
computing \cite{Clark07} may be generated from a few quantum gates, the
Hadamard, Phase and Controlled-Z gates, and efficiently simulated on a
classical computer. We employ the group theoretical package GAP\cite{GAP} for
simulating the two qubit Clifford group . We already found that
the symmetric group S(6), aka the automorphism group of the generalized
quadrangle W(2), controls the geometry of the two-qubit Pauli graph
\cite{Pauligraphs}. Now we find that the {\it inner} group
exactly contains
two normal subgroups, one isomorphic to (of order
16), and the second isomorphic to the parent (of order 5760) of the
alternating group A(6). The group stabilizes an {\it hexad} in the
Steiner system attached to the Mathieu group M(22). Both groups
A(6) and have an {\it outer} automorphism group , a feature we associate to two-qubit quantum entanglement.Comment: version for the journal Entrop
- …