No abstract

The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose i j-th entry (for i = j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. Maximum nullity is taken over the same set of matrices, and the sum of maximum nullity and minimum rank is the order of the graph. The zero forcing number is the minimum size of a zero forcing set of vertices and bounds the maximum nullity from above. This paper defines the graph families ciclos and estrellas and establishes the minimum rank and zero forcing number of several of these families. In particular, these families provide examples showing that the maximum nullity of a graph and its dual may differ, and similarly for the zero forcing number.

Abstract. The zero forcing number Z(G) is used to study the minimum rank/maximum nullity of the family of symmetric matrices described by a simple, undirected graph G. The positive semidefinite zero forcing number is a variant of the (standard) zero forcing number, which uses the same definition except with a different color-change rule. The positive semidefinite maximum nullity and zero forcing number for a variety of graph families are computed. In addition, field independence of the minimum rank of the hypercube is established, by showing there is a positive semidefinite matrix that is universally optimal.

Section 5.1: Area and Estimating with Finite Sums.• Describe upper and lower sums and what they tell you about area.• Given a partition and set {t 1 , . . . , t n } of points, with a single t i in each subinterval, draw the rectangles and calculate the associated Riemann Sum.

An oriented graph is a simple digraph obtained from a simple graph by choosing exactly one of the two arcs (u, v) or (v, u) to replace each edge {u, v}. A simple digraph describes the zero-nonzero pattern of off-diagonal entries of a family of (not necessarily symmetric) matrices. The minimum rank of a simple digraph is the minimum rank of this family of matrices; maximum nullity is defined analogously. The simple digraph zero forcing number and path cover number are related parameters. We establish bounds on the range of possible values of all these parameters for oriented graphs, establish connections between the values of these parameters for a simple graph G, for various orientations G and for the doubly directed digraph of G, and establish an upper bound on the number of arcs in a simple digraph in terms of the zero forcing number.