The Gram Dimension of a Graph

Abstract

The Gram dimension gd(G) of a graph is the smallest integer k ≥ 1 such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in R k , having the same inner products on the edges of the graph. The class of graphs satisfying gd(G) ≤ k is minor closed for fixed k, so it can characterized by a finite list of forbidden minors. For k ≤ 3, the only forbidden minor is K k+1 . We show that a graph has Gram dimension at most 4 if and only if it does not have K5 and K2,2,2 as minors. We also show some close connections to the notion of d-realizability of graphs. In particular, our result implies the characterization of 3-realizable graphs of Belk and Connelly [5,6].

Maximum Likelihood Threshold and Generic Completion Rank of Graphs

“…It is straightforward to see that both invariants are non-increasing with respect to edge and vertex deletion. However, unlike a related parameter of Gram dimension introduced in [22], the properties mlt(G) ≤ k and gcr(G) ≤ k are not minorclosed as the following example shows. Example 1.7.…”

Selected Open Problems in Discrete Geometry and Optimization

“…This parameter was introduced in [53,54] and its study is motivated by its connection with the low rank positive semidefinite matrix completion problem. In [53,54] it is shown that, for any fixed k 1, the class of graphs satisfying gd.G/ Ä k is closed under taking minors.…”

“…In [53,54] it is shown that, for any fixed k 1, the class of graphs satisfying gd.G/ Ä k is closed under taking minors. Moreover, it is shown that K kC1 is the only minimal forbidden minor for k 2 f1; 2; 3g and that K 5 and K 2;2;2 are the only minimal forbidden minors for the graph property gd.G/ Ä 4.…”

“…A first result in this direction was established in [53,54] where it was shown that, for any graph G,…”

“…This is motivated by an earlier result of So and Ye [20], which states that each dual variable in the SDP relaxation of Biswas and Ye [5] corresponds to a stress on an edge of the input graph, and the optimality conditions of the SDP correspond to a certain equilibrium condition on the input graph. The work [20] has since motivated or been used to develop other rigiditytheoretic results (see, e.g., [1,13]), and a natural question would be whether these results have counterparts in the Schatten quasi-norm regularization setting.…”