H. Tracy Hall

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.

Zero forcing in a graph refers to the evolution of vertex states under repeated application of a color change rule. Typically the states are chosen to be blue and white, and a forcing set is an initial set of blue vertices such that all of the vertices are blue at the end of the process. In this context, the propagation time of a set in a graph is the number of iterations of the color change rule required to have all vertices blue, performing independent color changes simultaneously. Different minimal forcing sets need not have the same propagation time, and we study the realizability of specific integers as propagation times of minimal forcing sets in graphs for two of the most well-studied color change rules (standard and positive semidefinite). Particular attention is paid to the case where all minimal forcing sets have the same propagation time, and we term this phenomenon fixed propagation time. For each of the two variants, we present a general form of graphs all of which have fixed propagation time equal to one. We conjecture that these are the only such graphs and prove the conjectures for joins of graphs. Families of graphs with longer fixed propagation time for standard forcing are exhibited, and it is shown that such graphs do not exist for positive semidefinite forcing.

The inverse eigenvalue problem of a graph studies the real symmetric matrices whose off-diagonal pattern is prescribed by the adjacencies of the graph. The strong spectral property (SSP) is an important tool for this problem. This note establishes the bifurcation lemma, which states that if a spectrum can be realized by a matrix with the SSP for some graph, then all the nearby spectra can also be realized by matrices with the SSP for the same graph. The idea of the bifurcation lemma also works for other strong properties and for not necessarily symmetric matrices. This is used to develop new techniques for verifying a spectrally arbitrary pattern or inertially arbitrary pattern. The bifurcation lemma provides a unified theoretical foundation for several known results, such as the stable northeast lemma and the nilpotent-centralizer method.