Spectrally extremal vertices, strong cospectrality and state transfer

TL;DR

This paper characterizes the structure of graphs with strongly cospectral vertices exhibiting extremal spectral properties, revealing conditions under which perfect state transfer occurs and linking it to distance-regular graphs.

Contribution

It provides a detailed structural analysis of graphs with strongly cospectral vertices and extremal spectral properties, connecting perfect state transfer to distance-regularity.

Findings

Graphs with extremal spectral properties and strong cospectrality are structurally characterized.

In regular graphs, perfect state transfer implies the graph is distance-regular.

Partitioning vertices at maximum distance with perfect state transfer leads to distance-regular graphs.

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 combinatorial 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.