Universal completability, least eigenvalue frameworks, and vector colorings

TL;DR

This paper explores universally completable graph embeddings derived from least eigenvalue eigenvectors, providing conditions for universality, algorithms for verification, and applications to graph coloring problems, including Kneser graphs.

Contribution

It establishes necessary and sufficient conditions for universal completability of least eigenvalue frameworks and connects this to uniquely vector colorable graphs, with new characterizations and algorithms.

Findings

Almost all Cayley graphs on ^n (n  5) have universally completable least eigenvalue frameworks.

Kneser and q-Kneser graphs are shown to be uniquely vector colorable.

Least eigenvalue frameworks of 1-walk-regular graphs always yield optimal vector colorings.

Abstract

An embedding $i \mapsto p_i\in \mathbb{R}^d$ of the vertices of a graph $G$ is called universally completable if the following holds: For any other embedding $i\mapsto q_i~\in \mathbb{R}^{k}$ satisfying $q_i^T q_j = p_i^T p_j$ for $i = j$ and $i$ adjacent to $j$, there exists an isometry mapping the $q_i$'s to the $p_i$'s for all $ i\in 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 $\mathbb{Z}_2^n \ (n \le 5)$ 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 Lov\'asz 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.