Wayne Barrett

Graphs whose minimal rank is two

Let F be a field, G =( V, E) be an undirected graph on n vertices, and let S(F, G) be the set of all symmetric n × n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. For example, if G is a path, S(F, G )co nsists of the symmetric irreducible tridiagonal matrices. Let mr(F, G) be the minimum rank over all matrices in S(F, G). Then mr(F, G) = 1 if and only if G is the union of a clique with at least 2 vertices and an independent set. If F is an infinite field such that charF � , then mr(F, G) ≤ 2i f and only if the complement of G is the join of a clique and a graph that is the union of at most two cliques and any number of complete bipartite graphs. A similar result is obtained in the case that F is an infinite field with char F = 2. Furthermore, in each case, such graphs are characterized as those for which 6 specific graphs do not occur as induced subgraphs. The number of forbidden subgraphs is reduced to 4 if the graph is connected. Finally, similar criteria is obtained for the minimum rank of a Hermitian matrix to be less than or equal to two. The complement is the join of a clique and a graph that is the union of any number of cliques and any number of complete bipartite graphs. The number of forbidden subgraphs is now 5, or in the connected case, 3.

A theorem on inverse of tridiagonal matrices

Abstract Tridiagonal or Jacobi matrices arise in many diverse branches of mathematics and have been studied extensively. However, there is little written about the inverses of such matrices. In this paper we characterize those matrices with nonzero diagonal elements whose inverses are tridiagonal. The arguments given are elementaryand show that matrices with tridiagonal inverses have an interesting structure.

Inverses of banded matrices

Abstract We establish a correspondence between the vanishing of a certain set of minors of a matrix A and the vanishing of a related set of minors of A ×1 . In particular, inverses of banded matrices are characterized. We then use our results to find patterns for Toeplitz matrices with banded inverses. Finally we give an interesting determinant formula for inverses of banded matrices, and show that in general a “banded partial” matrix may be completed in a unique way to give a banded inverse of the same bandwidth.

Determinantal formulae for matrix completions associated with chordal graphs

Abstract Let A be a partial positive definite Hermitian matrix, and let G ( A ) be the undirected graph of the specified off-diagonal entries of A . When G ( A ) is a chordal graph, we present an explicit determinantal formula for the (unique) determinant-maximizing positive definite completion (maximum-entropy completion) of A . In order to do this, we first give a characterization of a chordal graph in terms of the existence of a proper spanning tree T of the intersection graph of its set of maximal cliques. The formula for the maximum determinant is then a quotient of products of specified principal minors of A , in which each minor in the numerator corresponds to a maximal clique of G ( A ) while each minor in the denominator corresponds to the intersection of two maximal cliques associated with an edge of T . We next show that the denominator terms can be identified graph-theoretically with the minimal vertex separators of G ( A ), and that these terms are independent of the particular proper spanning tree T . Then an alternative formula is given for the maximum determinant as an “inclusion-exclusion”-like ratio of principal minors associated with the maximal cliques of G ( A ). Finally, we show that the entries of the determinant-maximizing completion may be calculated by solving a sequence of simple one-variable maximization problems.

Spanning Tree Extensions of the Hadamard-Fischer Inequalities

Abstract All possible graph-theoretic generalizations of a certain sort for the Hadamard-Fischer determinantal inequalities are determined. These involve ratios of products of principal minors which dominate the determinant. Furthermore, the cases of equality in these inequalities are characterized, and equality is possible for every set of values which can occur for the relevant minors. This relates recent work of the authors on positive definite completions and determinantal identities. When applied to the same collections of principal minors, earlier generalizations give poorer, more difficult to compute bounds than the present inequalities. Thus, this work extends, and in a certain sense completes, a series of generalizations of Hadamard-Fischer begun in the 1960s.

Determinantal formulae for matrices with sparse inverses

Abstract The determinant of a matrix is expressed in terms of certain of its principal minors by a formula which can be “read off” from the graph of the inverse of the matrix. The only information used is the zero pattern of the inverse, and each zero pattern yields one or more corresponding formulae for the determinant.

Graphs whose minimal rank is two : the finite fields case

Let F be a finite field, G =( V, E) be an undirected graph on n vertices, and let S(F, G) be the set of all symmetric n × n matrices over F whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G.L et mr(F, G) be the minimum rank of all matrices in S(F, G). If F is a finite field with p t elements, p � , it is shown that mr(F, G) ≤ 2i f and only if the complement of G is the join of a complete graph with either the union of at most (p t +1)/2 nonempty complete bipartite graphs or the union of at most two nonempty complete graphs and of at most (p t − 1)/2 nonempty complete bipartite graphs. These graphs are also characterized as those for which 9 specific graphs do not occur as induced subgraphs. If F is a finite field with 2t elements, then mr(F, G) ≤ 2 if and only if the complement of G is the join of a complete graph with either the union of at most 2t + 1 nonempty complete graphs or the union of at most one nonempty complete graph and of at most 2 t−1 nonempty complete bipartite graphs. A list of subgraphs that do not occur as induced subgraphs is provided for this case as well.

Determinantal inequalities for positive definite matrices

Abstract We consider the problem of identifying all determinantal inequalities valid on all positive definite matrices. This is fundamentally a combinatorial problem about relations between collections of index sets. We describe some general structure of this problem and give sufficient and necessary conditions that coincide for collections of no more than 3 index sets each.

Inertia sets for graphs on six or fewer vertices

Let G be an undirected graph on n vertices and let S(G) be the set of all real symmetric n×n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The inverse inertia problem for G asks which inertias can be attained by a matrix in S(G), a question which was previously answered when G is a tree. In this paper, a number of new techniques are developed in order to be able to determine possible inertias of general graphs: covers with cliques, covers with cliques and clique-stars, and the graph operations of edge subdivision, edge deletion, joins, and unions. Because most of the associated theorems require additional hypotheses, definitive criteria that apply to all graphs cannot be provided. Nevertheless, these results are strong enough to be able to determine the inertia set of each graph on 6 or fewer vertices and can be applied to many graphs with larger order as well. One consequence of the 1–6 vertex results is the fact that all of these graphs have balanced inertia. It is also mentioned which of these results guarantee or preserve balanced inertia, and explain how to modify them to include Hermitian matrices.

Minimum rank of edge subdivisions of graphs

Let F be a field, let G be an undirected graph on n vertices, and let S(F, G) be the set of all F -valued symmetric n × n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The minimum rank of G over F is defined to be mr(F, G) = min{rank A| A 2 S(F, G)}. The problem of finding the minimum rank (maximum nullity) of edge subdivisions of a given graph G is investigated. Is is shown that if an edge is adjacent to a vertex of degree 1 or 2, its maximum nullity is unchanged upon subdividing the edge. This enables us to reduce the problem of finding the minimum rank of any graph obtained from G by subdividing edges to finding the minimum rank of those graphs obtained from G by subdividing each edge at most once. The graph obtained by subdividing each edge of G once is called its subdivision graph and is denoted by a G. It is shown that its maximum nullity is an upper bound for the maximum nullity of any graph obtained from G by subdividing edges. It is also shown that the minimum rank of a G often depends only upon the number of vertices of G. In conclusion, some illustrative examples and open questions are presented.