The Gram Dimension of a Graph

  title={The Gram Dimension of a Graph},
  author={Monique Laurent and Antonios Varvitsiotis},
The Gram dimension $\text{\rm gd}(G)$ 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 ℝk, having the same inner products on the edges of the graph. The class of graphs satisfying $\text{\rm gd}(G) \le k$ 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 Kk+1. We show that a graph has Gram… 
A new graph parameter related to bounded rank positive semidefinite matrix completions
The Gram dimension $$\mathrm{gd}(G)$$ of a graph $$G$$ is the smallest integer $$k\ge 1$$ such that any partial real symmetric matrix, whose entries are specified on the diagonal and at the
Approximate PSD-Completion for Generalized Chordal Graphs
Recently, there has been interest in the question of whether a partial matrix in which many of the fully defined principal submatrices are PSD is approximately PSD completable. These questions are
Maximum Likelihood Threshold and Generic Completion Rank of Graphs
This work determines both invariants for complete bipartite graphs Km,n and shows that for some choices of m and n the two parameters may be quite far apart, which gives the first examples of graphs on which the maximum likelihood threshold and the generic completion rank do not agree.
Combinatorial conditions for low rank solutions in semidefinite programming
This thesis investigates combinatorial conditions that guarantee the existence of low-rank optimal solutions to semidefinite programs and introduces a graph parameter called the extreme Gram dimension of a graph, which is upper bounded by the Gram dimension and closely related to the rank-constrained Grothendieck constant.
Iterative Universal Rigidity
This work provides a characterization of universal rigidity for any graph G and any configuration p in terms of a sequence of affine subsets of the space of configurations, which corresponds to a facial reduction process for closed finite-dimensional convex cones.
Complexity of the positive semidefinite matrix completion problem with a rank constraint
We consider the decision problem asking whether a partial rational symmetric matrix with an all-ones diagonal can be completed to a full positive semidefinite matrix of rank at most k. We show that
Extreme Nonnegative Quadratics over Stanley Reisner Varieties
We consider the convex geometry of the cone of nonnegative quadratics over Stanley-Reisner varieties. Stanley-Reisner varieties (which are unions of coordinate planes) are amongst the simplest real
Discrete geometry and optimization
Preface.- Discrete Geometry in Minkowski Spaces (Alonso, Martini, and Spirova).- Engineering Branch-and-Cut Algorithms for the Equicut Program (Anjos, Liers, Pardella, and Schmutzer).- An Approach to
A bound on the minimum rank of solutions to sparse linear matrix equations
We derive a new upper bound on the minimum rank of matrices belonging to an affine slice of the positive semidefinite cone, when the affine slice is defined according to a system of sparse linear


A new graph parameter related to bounded rank positive semidefinite matrix completions
The Gram dimension $$\mathrm{gd}(G)$$ of a graph $$G$$ is the smallest integer $$k\ge 1$$ such that any partial real symmetric matrix, whose entries are specified on the diagonal and at the
Graph Minors
For a given graph G and integers b, f ≥ 0, let S be a subset of vertices of G of size b + 1 such that the subgraph of G induced by S is connected and S can be separated from other vertices of G by
Orthogonal representations over finite fields and the chromatic number of graphs
It turns out that for some classes of matrices defined by a graph the 3-colorability problem is equivalent to deciding whether the class defined by the graph contains a matrix of rank 3 or not, which implies the NP-hardness of determining the minimum rank of a matrix in such a class.
On the Shannon capacity of a graph
  • L. Lovász
  • Mathematics, Computer Science
    IEEE Trans. Inf. Theory
  • 1979
It is proved that the Shannon zero-error capacity of the pentagon is \sqrt{5} and a well-characterized, and in a sense easily computable, function is introduced which bounds the capacity from above and equals the capacity in a large number of cases.
Realizability of Graphs in Three Dimensions
The two graphs V8 and C5 × C2 are shown to be 3-realizable, which means that the forbidden minors for 3- realizability are K5 and K2, 2,2.
Realizability of Graphs
It is shown that a graph is 3-realizable if and only if it does not have K5 or the 1-skeleton of the octahedron as a minor.
Forbidden minors characterization of partial 3-trees
Orthogonal representations, minimum rank, and graph complements
A semidefinite programming approach to tensegrity theory and realizability of graphs
This paper uses SDP duality theory to show that given a graph G and a set of lengths on its edges, the optimal dual multipliers of a certain SDP give rise to a proper equilibrium stress for some realization of G, and obtains a polynomial time algorithm for realizing 3-realizable graphs.
Geometry of cuts and metrics
This book draws from the interdisciplinarity of these fields as it gathers methods and results from polytope theory, geometry of numbers, probability theory, design and graph theory around two objects, cuts and metrics.