Universal Completability, Least Eigenvalue Frameworks, and Vector Colorings
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F17%3A10369131" target="_blank" >RIV/00216208:11320/17:10369131 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1007/s00454-017-9899-2" target="_blank" >http://dx.doi.org/10.1007/s00454-017-9899-2</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s00454-017-9899-2" target="_blank" >10.1007/s00454-017-9899-2</a>
Alternative languages
Result language
angličtina
Original language name
Universal Completability, Least Eigenvalue Frameworks, and Vector Colorings
Original language description
An embedding $i mapsto p_iin R^d$ of the vertices of a graph $G$ is called universally completable if the following holds: For any other embedding $imapsto q_i~in R^{k}$ satisfying $q_itranspose q_j = p_itranspose p_j$ for $i = j$ and $i$ adjacent to $j$, there exists an isometry mapping the $q_i$'s to the $p_i$'s for all $ iin V(G)$. The notion of universal completability was introduced recently due to its relevance to the positive semidefinite matrix completion problem. In this work we focus on graph embeddings constructed using the eigenvectors of the least eigenvalue of the adjacency matrix of $G$, which we call least eigenvalue frameworks. We identify two necessary and sufficient conditions for such frameworks to be universally completable. Our conditions also allow us to give algorithms for determining whether a least eigenvalue framework is universally completable. Furthermore, our computations for Cayley graphs on $Z_2^n (n le 5)$ show that almost all of these graphs have universally completable least eigenvalue frameworks. In the second part of this work we study uniquely vector colorable (UVC) graphs, i.e., graphs for which the semidefinite program corresponding to the Lovász theta number (of the complementary graph) admits a unique optimal solution. We identify a sufficient condition for showing that a graph is UVC based on the universal completability of an associated framework. This allows us to prove that Kneser and $q$-Kneser graphs are UVC. Lastly, we show that least eigenvalue frameworks of 1-walk-regular graphs always provide optimal vector colorings and furthermore, we are able to characterize all optimal vector colorings of such graphs. In particular, we give a necessary and sufficient condition for a 1-walk-regular graph to be uniquely vector~colorable.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Result continuities
Project
Result was created during the realization of more than one project. More information in the Projects tab.
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2017
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Discrete and Computational Geometry
ISSN
0179-5376
e-ISSN
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Volume of the periodical
58
Issue of the periodical within the volume
2
Country of publishing house
US - UNITED STATES
Number of pages
28
Pages from-to
265-292
UT code for WoS article
000406409600002
EID of the result in the Scopus database
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