Geodesic
Spectrally extremal vertices, strong cospectrality, and state transfer
Gabriel Coutinho1
1University of Waterloo
The electronic journal of combinatorics, Tome 23 (2016) no. 1
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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.
DOI : 10.37236/5031
Classification : 05C50, 05E30, 81P68
Mots-clés : graph theory, quantum walks, cospectral vertices

Gabriel Coutinho  1

1 University of Waterloo 
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     author = {Gabriel Coutinho},
     title = {Spectrally extremal vertices, strong cospectrality, and state transfer},
     journal = {The electronic journal of combinatorics},
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Gabriel Coutinho. Spectrally extremal vertices, strong cospectrality, and state transfer. The electronic journal of combinatorics, Tome 23 (2016) no. 1. doi: 10.37236/5031

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