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Entanglement Theory Quantum Correlations

Tales of Hoffman: from a distance

arXiv
Authors: Aida Abiad, Jan Meeus

Year

2025

Paper ID

36539

Status

Preprint

Abstract Read

~2 min

Abstract Words

156

Citations

N/A

Abstract

Hoffman proved that a graph G with adjacency eigenvalues λ1geq cdots geq λn and chromatic number χ(G) satisfies χ(G)geq 1+κ, where κ is the smallest integer such that $λ1+sumi=1^κλn+1-ileq 0.We extend this eigenvalue bound to the distance-ksetting, and also show a strengthening of it by proving that it also lower bounds the corresponding quantum distance coloring graph parameter. The new bound depends on a degree-k$ polynomial which can be chosen freely, so one needs to make a good choice of the polynomial to obtain as strong a bound as possible. We thus propose linear programming methods to optimize it. We also investigate the implications of the new bound for the quantum distance chromatic number, showing that it is sharp for some classes of graphs. Finally, we extend the Hoffman bound to the distance setting of the vector chromatic number. Our results extend and unify several previous bounds in the literature.

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  • This paper contributes to the Entanglement Theory & Quantum Correlations research area in the Quantum Articles archive.
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  • Hoffman proved that a graph G with adjacency eigenvalues λ1geq cdots geq λn and chromatic number χ(G) satisfies χ(G)geq 1+κ, where κ is the smallest integer such that...

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References & Citation Signals

[1] DOI https://doi.org/arXiv:2512.13187 [2] arXiv https://arxiv.org/abs/2512.13187 [3] Publisher https://arxiv.org/abs/2512.13187

Local Citation Graph (Related-Paper Links)

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