Abstract
In order to obtain perfect state transfer between two sites in a network of interacting qubits, their corresponding vertices in the underlying graph must satisfy a property called strong cospectrality. Here we determine the structure of graphs containing pairs of vertices which are strongly cospectral and satisfy a certain extremal property related to the spectrum of the graph. If the graph satisfies this property globally and is regular, we also show that the existence of a partition of the vertex set into pairs of vertices at maximum distance admitting perfect state transfer forces the graph to be distance-regular. Finally, we present some new examples of perfect state transfer in simple graphs constructed with our technology. In particular, for odd distances, we improve the known trade-off between the distance perfect state transfer occurs in simple graphs and the size of the graph.
Highlights
Let X be a simple undirected graph, and consider its 01-adjacency matrix A = A(X)
If the graph satisfies this property globally and is regular, we show that the existence of a partition of the vertex set into pairs of vertices at maximum distance admitting perfect state transfer forces the graph to be distance-regular
We present some new examples of perfect state transfer in simple graphs constructed with our technology
Summary
We will examine this property in the context of graphs which are extremal with respect to the lower bound on the number of eigenvalues given by the diameter plus one State transfer on such graphs was considered in [21]. We will consider the extremal spectral property locally, and show that a pair of strongly cospectral extremal vertices at maximum distance must be singletons in a pseudo equitable distance partition This will be our key intermediate step to show that if an extremal regular graph of diameter d can be partitioned into pairs of vertices at distance d such that perfect state transfer happens in each pair, the graph is distance-regular. In the context of distance-regular graphs, this will provide an alternate elementary proof of [8, Corollary 4.5]
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