The Delta Theorem: a dimension bound for faithful orthogonal graph representations

Abstract

In 1987 Hiroshi Maehara conjectured that a graph can be represented by vectors considered adjacent when not orthogonal (a faithful orthogonal representation) in codimension the minimum degree of the graph. Without settling the conjecture, Làslò Lovàsz, Michael Saks, and Alexander Schrijver (LSS) showed that a codimension of vertex connectivity both suffices and is best possible under the additional assumption of general position, and gave a probabilistic construction for producing such representations. The present work proves the conjecture of Maehara as well as related conjectures, variants of the Delta Conjecture, that have arisen independently in combinatorial matrix theory. The strongest of these is that minimum degree of G gives a lower bound for the maximum nullity of a positive definite matrix with pattern G that has the Strong Arnold Property (SAP). Such nullity questions are an important subcase of the Inverse Eigenvalue Problem for a Graph (IEPG). The name greedegree is introduced for the largest possible final degree of a maximum cardinality search (MSC) ordering, which is to say an ordering that greedily maximizes adjacencies to previous chosen vertices. The name upper-zero generic is introduced to describe symmetric matrices with nonzero diagonal such that the zeros above the diagonal in any column belong to an independent set of rows, which matrices necessarily have the SAP. The proof technique takes the probabilistic construction of LSS and parametrizes it completely in terms of independent variables, producing large polynomials that are reasoned about using an introduced operad of hanging garden diagrams. In the case of an MSC ordering in codimension greedegree, it is shown that the leading monomial in an appropriate term order has no canceling term, giving a nonzero polynomial. The resulting representation is faithful with upper-zero generic Gram matrix.