Distribution of embeddings

Groß, Joachim; Tucker, Thomas W.

doi:10.1017/cbo9781139087223.006

Cited by 217 publications

(402 citation statements)

References 0 publications

Supporting

Mentioning

398

Contrasting

Order By: Relevance

“…Cayley graphs are defined in terms of a group G and a set S of elements from G, chosen such that the identity element e / ∈ S. Given G and S, the resulting (right)-Cayley graph Γ(G, S) is one whose vertices are labeled by the group elements and whose (edge) directions are labeled by the elements of S. There is one vertex for every group element, and two vertices g and h are connected by a directed edge from g to h if g −1 h ∈ S, (see [32]). Another way to look at this definition is that from any vertex g of a Cayley graph, there are |S| outgoing edges, one to each of the vertices gs, ∀s ∈ S. A Cayley graph will be connected if and only if the set S is a generating set for G, it will be undirected if s −1 ∈ S, ∀s ∈ S and it will be d-colorable if s −1 = s, ∀s ∈ S. Cayley graphs are always regular, and the degree of a Cayley graph is |S|, the cardinality of the generating set.…”

Section: Cayley Graphs and Automorphism Groupsmentioning

confidence: 99%

Quantum walks on quotient graphs

2007

View full text Add to dashboard Cite

A discrete-time quantum walk on a graph Γ is the repeated application of a unitary evolution operator to a Hilbert space corresponding to the graph. If this unitary evolution operator has an associated group of symmetries, then for certain initial states the walk will be confined to a subspace of the original Hilbert space. Symmetries of the original graph, given by its automorphism group, can be inherited by the evolution operator. We show that a quantum walk confined to the subspace corresponding to this symmetry group can be seen as a different quantum walk on a smaller quotient graph. We give an explicit construction of the quotient graph for any subgroup H of the automorphism group and illustrate it with examples. The automorphisms of the quotient graph which are inherited from the original graph are the original automorphism group modulo the subgroup H used to construct it. The quotient graph is constructed by removing the symmetries of the subgroup H from the original graph. We then analyze the behavior of hitting times on quotient graphs. Hitting time is the average time it takes a walk to reach a given final vertex from a given initial vertex. It has been shown in earlier work [Phys. Rev. A 74, 042334 (2006)] that the hitting time for certain initial states of a quantum walks can be infinite, in contrast to classical random walks. We give a condition which determines whether the quotient graph has infinite hitting times given that they exist in the original graph. We apply this condition for the examples discussed and determine which quotient graphs have infinite hitting times. All known examples of quantum walks with hitting times which are short compared to classical random walks correspond to systems with quotient graphs much smaller than the original graph; we conjecture that the existence of a small quotient graph with finite hitting times is necessary for a walk to exhibit a quantum speed-up.

show abstract

Section: Cayley Graphs and Automorphism Groupsmentioning

confidence: 99%

Quantum walks on quotient graphs

2007

View full text Add to dashboard Cite

show abstract

“…Moreover, the action of A on the vertex set is left multiplication, so the cyclic ordering of generators X and their inverses must be the same at every vertex of the map. Thus, a Cayley map for the group A is a strongly symmetric embedding of a Cayley graph for A, in the language of [23], or the derived graph of a one-vertex voltage graph with a directed edge for each generator in X, in the language of [9]. For a detailed development of the theory of Cayley maps, including their recognition from a combinatorial description of a map, their relationship to planar tessellations, and their role as universal regular coverings, we cite [18], which forms much of the motivation for this paper.…”

Section: Introductionmentioning

confidence: 99%

Regular Cayley maps for finite abelian groups

2006

View full text Add to dashboard Cite

A regular Cayley map for a finite group A is an orientable map whose orientation-preserving automorphism group G acts regularly on the directed edge set and has a subgroup isomorphic to A that acts regularly on the vertex set. This paper considers the problem of determining which abelian groups have regular Cayley maps. The analysis is purely algebraic, involving the structure of the canonical form for A. The case when A is normal in G involves the relationship between the rank of A and the exponent of the automorphism group of A, and the general case uses Ito's theorem to analyze the factorization G = AY , where Y is the (cyclic) stabilizer of a vertex.

show abstract

“…Such an assignment of cyclic orderings constituted a combinatorial planar embedding of the tree (and is called a rotation system; see [1]). Several application areas involve the comparison between planar embedded trees.…”

Section: Introductionmentioning

confidence: 99%

Computing the Edit-Distance Between Unrooted Ordered Trees

Klein

1998

Algorithms — ESA’ 98

177

175

View full text Add to dashboard Cite

Abstract. An ordered tree is a tree in which each node's incident edges are cyclically ordered; think of the tree as being embedded in the plane. Let A and B be two ordered trees. The edit distance between A and B is the minimum cost of a sequence of operations (contract an edge, uncontract an edge, modify the label of an edge) needed to transform A into B. We give an O(n 3 log n) algorithm to compute the edit distance between two ordered trees.

show abstract

Distribution of embeddings

Cited by 217 publications

References 0 publications

Quantum walks on quotient graphs

Quantum walks on quotient graphs

Regular Cayley maps for finite abelian groups

Computing the Edit-Distance Between Unrooted Ordered Trees

Contact Info

Product

Resources

About