2017/09/23 by Chris Godsil, Jamie Smith, Godsil, Chris +1 · 3 citations
Computer Science · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Mathematical Approximation and Integration #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.1709.07975
openalex publication_date 2017/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Two vertices a and b in a graph X are cospectral if the vertex-deleted subgraphs X∖ a and X∖ b have the same characteristic polynomial. In this paper we investigate a strengthening of this relation on vertices, that arises in investigations of continuous quantum walks. Suppose the vectors ea for a in V(X) are the standard basis for ℝV(X). We say that a and b are strongly cospectral if, for each eigenspace U of A(X), the orthogonal projections of ea and eb are either equal or differ only in sign. We develop the basic theory of this concept and provide constructions of graphs with pairs of strongly cospectral vertices. Given a continuous quantum walk on on a graph, each vertex determines a curve in complex projective space. We derive results that show tht the closer these curves are, the more "similar" the corresponding vertices are.