Francesco Barioli

A variant on the graph parameters of Colin de Verdiere: Implications to the minimum rank of graphs

For a given undirected graph G, the minimum rankof G is defined to be the smallest possible rankover all real symmetric matrices A whose (i, j)th entry is nonzero whenever ij and {i, j} is an edge in G. Building upon recent workinvolving maximal corank s (or nu llities) of certain symmetric matrices associated with a graph, a new parameter ξ is introduced that is based on the corankof a different but related class of symmetric matrices. For this new parameter some properties analogous to the ones possessed by the existing parameters are verified. In addition, an attempt is made to apply these properties associated with ξ to learn more about the minimum rankof graphs - the original motivation.

ON THE MINIMUM RANK OF NOT NECESSARILY SYMMETRIC MATRICES: A PRELIMINARY STUDY ∗

The minimum rank of a directed graph Γ is defined to be the smallest possible rank over all real matrices whose ijth entry is nonzero whenever (i,j) is an arc in Γ and is zero otherwise. The symmetric minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for ij) is nonzero whenever {i,j} is an edge in G and is zero otherwise. Maximum nullity is equal to the difference between the order of the graph and minimum rank in either case. Definitions of various graph parameters used to bound symmetric maximum nullity, including path cover number and zero forcing number, are extended to digraphs, and additional parameters related to minimum rank are introduced. It is shown that for directed trees, maximum nullity, path cover number, and zero forcing number are equal, providing a method to compute minimum rank for directed trees. It is shown that the minimum rank problem for any given digraph or zero-nonzero pattern may be converted into a symmetric minimum rank problem.

The maximal cp-rank of rank k completely positive matrices

Abstract Let Φ k be the maximal cp-rank of all rank k completely positive matrices. We prove that Φ k = k ( k +1)/2−1 for k ⩾2. In particular we furnish a procedure to produce, for k ⩾2, completely positive matrices with rank k and cp-rank k ( k +1)/2−1.

ON TWO CONJECTURES REGARDING AN INVERSE EIGENVALUE PROBLEM FOR ACYCLIC SYMMETRIC MATRICES

For a given acyclic graph G, an important problem is to characterize all of the eigenvalues over all symmetric matrices with graph G. Of particular interest is the connection between this standard inverse eigenvalue problem and describing all the possible associated ordered multiplicity lists, along with determining the minimum number of distinct eigenvalues for a symmetric matrix with graph G. In this note two important open questions along these lines are resolved, both in the negative.

On the eigenvalues of generalized and double generalized stars

We consider describing all possible spectra of symmetric matrices associated with certain graphs by characterizing all possible ordered multiplicity lists. For generalized and double generalized stars we provide a complete description of all possible eigenvalue sequences. This result is obtained by first verifying all possible multiplicity lists for these graphs which, in fact, turns out to be necessary and sufficient for determining all eigenvalue sequences corresponding to these graphs.

COMPLETELY POSITIVE MATRICES WITH A BOOK-GRAPH

Abstract A new class of graphs, called “book-graphs”, extending the class of completely positive graphs is defined. Necessary and sufficient conditions for the complete positivity of a matrix with graph in this class are given. The main questions concerning completely positive matrices with cyclic graph are solved.

Chains of dog-ears for completely positive matrices

Abstract A necessary and sufficient condition to determine the complete positivity of a matrixwith a particular graph, in dependence of complete positivity of smaller matrices, is given. Under some singularity assumptions, this condition furnishes a characterization for completely positive matrices with a “non-crossing cycle” as associated graph. In particular the characterization holds for singular pentadiagonal matrices.

On vector spaces with distinguished subspaces and redundant base

Abstract Let V , W be finite dimensional vector spaces over a field K , each with n distinguished subspaces, with a dimension-preserving correspondence between intersections. When does this guarantee the existence of an isomorphism between V and W matching corresponding subspaces? The setting where it happens requires that the distinguished subspaces be generated by subsets of a given redundant base of the space; this gives rise to a (0,1)-incidence table called tent , an object which occurs in the study of Butler B (1)-groups.