Spectrally extremal vertices, strong cospectrality, and state transfer

• Autor(a) principal: Gabriel de Morais Coutinho
• Data de Publicação: 2016
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Institucional da UFMG
• Texto Completo: https://doi.org/10.37236/5031; http://hdl.handle.net/1843/38122; https://orcid.org/0000-0002-1581-431X
• topic: Graph theory; Quantum walks; Cospectral vertices; Teoria dos grafos; Análise combinatória; Análise espectral; Computação quântica

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.