Abstract
An embedding i mapsto p_iin mathbb {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 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 q_i to p_i 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 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 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.
Highlights
A tensegrity graph is defined as a graph G = ([n], E) where the edge set E is partitioned into three disjoint sets B, C, and S
In this paper we address this problem by using least eigenvalue frameworks (LEF), i.e., graph embeddings obtained using the eigenvectors corresponding to the least eigenvalue of the adjacency matrix of the graph
In the second part of this paper we focus on uniquely vector colorable (UVC) graphs, i.e., graphs for which one of the semidefinite programming formulations corresponding to the Lovász theta number of the complementary graph admits a unique optimal solution
Summary
For the remainder of this section we consider the special case where both S and C in the definition of S(G, p) are empty In this setting, any tensegrity framework G(p) defines a G-partial matrix, i.e., a matrix whose entries are only specified along the diagonal and off-diagonal positions corresponding to edges of the G. The Gram dimension of a graph G, denoted by gd(G), was introduced in [20] to address the low-rank psd matrix completion problem described above. It is defined as the smallest integer k ≥ 1 with the following property: For any framework G(p) there exists an element Z ∈ S(G, p) such that rank(Z ) ≤ k. For a detailed comparison between these two notions the reader is referred to [19]
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