Ground-State Entanglement in Interacting Bosonic Graphs

We consider a collection of bosonic modes corresponding to the vertices of a graph $\Gamma.$ Quantum tunneling can occur only along the edges of $\Gamma$ 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 $\tau.$ The topology of $\Gamma$ plays a major role in determining the tunneling amplitude $\tau^*$ which leads to the maximum ground-state entanglement. Whereas in most of the cases one finds the intuitively expected result $\tau^*=\infty$ we show that it there exists a family of graphs for which the optimal value of$\tau$ 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