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Universal Completability, Least Eigenvalue Frameworks, and Vector Colorings.

Chris Godsil1, David E Roberson2, Brendan Rooney3

  • 11Department of Combinatorics & Optimization, University of Waterloo, Waterloo, ON N2L 3G1 Canada.

Discrete & Computational Geometry
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Summary
This summary is machine-generated.

This study introduces universally completable graph embeddings and least eigenvalue frameworks, providing conditions and algorithms for their analysis. It also proves Kneser and q-Kneser graphs are uniquely vector colorable (UVC).

Keywords:
Least eigenvalueLovász theta numberPositive semidefinite matrix completionSemidefinite programmingUniversal rigidityVector colorings

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Area of Science:

  • Graph theory
  • Linear algebra
  • Combinatorics

Background:

  • Introduces universally completable graph embeddings, relevant to positive semidefinite matrix completion.
  • Focuses on least eigenvalue frameworks derived from graph adjacency matrix eigenvectors.
  • Defines uniquely vector colorable (UVC) graphs based on semidefinite programming for Lovász theta number.

Purpose of the Study:

  • To identify necessary and sufficient conditions for universally completable least eigenvalue frameworks.
  • To develop algorithms for determining universal completability of these frameworks.
  • To establish conditions for graphs to be UVC using associated frameworks and prove Kneser/q-Kneser graphs are UVC.

Main Methods:

  • Analysis of graph embeddings and their properties under isometric mappings.
  • Development of conditions based on graph structure and eigenvalues.
  • Application of these conditions to specific graph classes like Cayley, Kneser, and q-Kneser graphs.

Main Results:

  • Identified two necessary and sufficient conditions for universally completable least eigenvalue frameworks.
  • Showed that almost all Cayley graphs on n vertices have universally completable least eigenvalue frameworks.
  • Proved Kneser and q-Kneser graphs are UVC and characterized optimal vector colorings for 1-walk-regular graphs.

Conclusions:

  • Least eigenvalue frameworks offer a powerful tool for studying graph embeddings and matrix completion.
  • Universal completability provides a key property for understanding graph structure and unique vector colorability.
  • The study advances the understanding of UVC graphs and optimal vector colorings in graph theory.