Survey of Double Vertex Graphs

Alavi, Yousef; Lick, Don R.; Liu, Jiuqiang

doi:10.1007/s003730200055

Cited by 28 publications

(42 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The n-fold graph wedge product of a graph G, namely G n = n G, has been studied in the graph theory literature where it has been referred to as an n-tuple vertex graph 16,17 . There appears to be no literature on the spectral properties of n-tuple vertex graphs in general: the results of this paper appear to provide the first investigation of the spectral properties of such graphs.…”

mentioning

confidence: 99%

Statics and dynamics of quantumXYand Heisenberg systems on graphs

Osborne¹

2006

Phys. Rev. B

View full text Add to dashboard Cite

We consider the statics and dynamics of distinguishable spin-1/2 systems on an arbitrary graph G with N vertices. In particular, we consider systems of quantum spins evolving according to one of two hamiltonians: (i) the XY hamiltonian HXY , which contains an XY interaction for every pair of spins connected by an edge in G; and (ii) the Heisenberg hamiltonian HHeis, which contains a Heisenberg interaction term for every pair of spins connected by an edge in G. We find that the action of the XY (respectively, Heisenberg) hamiltonian on state space is equivalent to the action of the adjacency matrix (respectively, combinatorial laplacian) of a sequence G k , k = 0, . . . , N of graphs derived from G (with G1 = G). This equivalence of actions demonstrates that the dynamics of these two models is the same as the evolution of a free particle hopping on the graphs G k . Thus we show how to replace the complicated dynamics of the original spin model with simpler dynamics on a more complicated geometry. A simple corollary of our approach allows us to write an explicit spectral decomposition of the XY model in a magnetic field on the path consisting of N vertices. We also use our approach to utilise results from spectral graph theory to solve new spin models: the XY model and heisenberg model in a magnetic field on the complete graph. The techniques developed to solve interacting manybody quantum systems have led to the discovery of many new intriguing nonclassical phenomena. A canonical example is the discovery of quantum phase transitions 5,6 , phase transitions which occur in the ground state -a pure state -which are driven by quantum rather than thermal fluctuations. However, these techniques can typically only be applied to systems which possess a great deal of symmetry. Hence it is extremely desirable to develop new approaches that can be applied in more general situations.There is a superficial similarity between the mathematics of distinguishable quantum spins and the spectral theory of graphs 7,8,9 , which pertains to the dynamics of a single quantum particle hopping on a discrete graph. In both cases there is a graph structure and a notion of locality. In the case of graphs, locality can be characterised by the support of the particle wavefunction, i.e., the position of the quantum particle. A localised particle remains, for small times, approximately localised. (There is a natural UV cutoff given by the graph structure, hence there is a resulting bound on the propagation speed of the particle.) In the case of spin systems the notion of locality emerges in the Heisenberg picture where the support of operators takes the role of defining local physics. Under dynamics local operators remain approximately local for short times. (This locality result is a consequence of the Lieb-Robinson bound 10,11,12,13 .) In both the case of quantum spin systems and spectral graph theory we are interested in the eigenvalues and the eigenvectors of the generator of time translations: the hamiltonian for the spin system; and the ad...

show abstract

mentioning

confidence: 99%

Statics and dynamics of quantumXYand Heisenberg systems on graphs

Osborne¹

2006

Phys. Rev. B

View full text Add to dashboard Cite

show abstract

“…It is easy to see that the sequences of vertices (x, z) · · · (x, n) (2, n) (3, n) (3,2)(4, 2) · · · (n, 2)(x, 2) · · · (x, y)…”

Section: Andmentioning

confidence: 98%

“…In particular, paper [3] reviews the recent results. In this paper, we extend the definition of the double vertex graph of a graph to a directed graph, and establish some relationships between a directed graph and its double vertex directed graph.…”

Section: Introductionmentioning

confidence: 98%

“…For a simple graph G (with no loops and multiple edges), in [1] the authors gave the following definition. The double vertex graph U 2 (G) of G is the graph whose vertex set consists of all 2-subsets of V (G) such that two distinct vertices (x, y) and (u, v) are adjacent if and only if |{x, y} ∩ {u, v}| = 1 and if x = u, then y and v are adjacent in G. The order and the size of U 2 (G) are n(n − 1)/2 and q(n − 2), respectively, where n is the order and q is the size of G. As examples, we have U 2 (K 2 ) = K 1 , U 2 (K 3 ) = K 3 , and U 2 (K 1,3 ) = C 6 .…”

Section: Introductionmentioning

confidence: 99%

“…Recently, there have been some papers concerning this topic, for example [1][2][3]5,7]. In particular, paper [3] reviews the recent results.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Double vertex digraphs of digraphs

Gao

Shao

2009

Discrete Mathematics

View full text Add to dashboard Cite

a b s t r a c tLet D be a digraph of order n. The double vertex digraph S 2 (D) of D is the digraph whose vertex set consists of all ordered pairs of distinct vertices of V (D) such that there is an arc in S 2 (D) from (x, y) to (u, v) if and only if x = u and there is an arc in D from y to v, or y = v and there is an arc in D from x to u. In this paper, we establish some relationships between a digraph and its double vertex digraph.

show abstract

Some Results on the Laplacian Spectra of Token Graphs

Dalfó

Duque

Fabila-Monroy

et al. 2021

Trends in Mathematics

View full text Add to dashboard Cite

We study the Laplacian spectrum of token graphs, also called symmetric powers of graphs. The k-token graph F k (G) of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in G. In this work, we give a relationship between the Laplacian spectra of any two token graphs of a given graph. In particular, we show that, for any integers h and k such that 1 ≤ h ≤ k ≤ n 2 , the Laplacian spectrum of F h (G) is contained in the Laplacian spectrum of F k (G). Besides, we obtain a relationship between the spectra of the k-token graph of G and the k-token graph of its complement G. This generalizes a well-known property for Laplacian eigenvalues of graphs to token graphs.

show abstract

Survey of Double Vertex Graphs

Cited by 28 publications

References 1 publication

Statics and dynamics of quantumXYand Heisenberg systems on graphs

Statics and dynamics of quantumXYand Heisenberg systems on graphs

Double vertex digraphs of digraphs

Some Results on the Laplacian Spectra of Token Graphs

Contact Info

Product

Resources

About