14 research outputs found
Ground-State Entanglement in Interacting Bosonic Graphs
We consider a collection of bosonic modes corresponding to the vertices of a
graph Quantum tunneling can occur only along the edges of
and a local self-interaction term is present. Quantum entanglement of one
vertex with respect the rest of the graph is analyzed in the ground-state of
the system as a function of the tunneling amplitude The topology of
plays a major role in determining the tunneling amplitude
which leads to the maximum ground-state entanglement. Whereas in most of the
cases one finds the intuitively expected result we show that it
there exists a family of graphs for which the optimal value of is pushed
down to a finite value. We also show that, for complete graphs, our bi-partite
entanglement provides useful insights in the analysis of the cross-over between
insulating and superfluid ground statesComment: 5 pages (LaTeX) 5 eps figures include
Distance-regularised graphs are distance-regular or distance-biregular
One problem with the theory of distance-regular graphs is that it does not apply directly to the graphs of generalised polygons. In this paper we overcome this difficulty by introducing the class of distance-regularised graphs, a natural common generalisation. These graphs are shown to either be distance-regular or fall into a family of bipartite graphs called distance-biregular. This family includes the generalised polygons and other interesting graphs. Despite this increased generality we are also able to extend much of the basic theory of distance-regular graphs to our wider class of graphs
A permutation group determined by an ordered set
Abstract. Let P be a finite ordered set, and let J(P) be the distributive lattice of order ideals of P. The covering relations of J(P) are naturally associated with elements of P; in this way, each element of P defines an involution on the set J(P). Let Γ(P) be the permutation group generated by these involutions. We show that if P is connected then Γ(P) is either the alternating or the symmetric group. We also address the computational complexity of determining which case occurs. Let P be a finite ordered set, and let J(P) be the distributive lattice of order ideals (also called down–sets) of P. For each p ∈ P, define a permutation σp on J(P) as follows: for every S ∈ J(P), ⎨ S ∪ {p} if p is minimal in P � S, σp(S): = S � {p} S if p is maximal in S, otherwise. Each of these permutations is an involution. We let Γ(P) denote the subgroup of the symmetric group Sym(J(P)) generated by all these involutions. Plain curiosity led us to wonder about the structure of these permutation groups. As we shall see, this can be determined quite precisely. As an example, for P = c ���� � d a b we may number the down–sets {∅, a, b, ab, bd, abc, abd, abcd} of P by 1 through 8, and the
A permutation group determined by an ordered set
Let P be a finite ordered set, and let J(P) be the distributive lattice of order ideals of P. The covering relations of J(P) are naturally associated with elements of P; in this way, each element of P defines an involution on the set J(P). Let Γ(P) be the permutation group generated by these involutions. We show that if P is connected then Γ(P) is either the alternating or the symmetric group. We also address the computational complexity of determining which case occurs
Representations of directed strongly regular graphs
AbstractWe develop a theory of representations in Rm for directed strongly regular graphs, which gives a new proof of a nonexistence condition of Jørgensen [L.K. Jørgensen, Non-existence of directed strongly regular graphs, Discrete Math. 264 (2003) 111–126]. We also describe some new constructions