For a graph G of order n, the minimum rank of G is defined to be the smallest possible rank over all real symmetric n × n matrices A whose (i, j)th entry (for i 6= j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. We prove an upper bound for minimum rank in terms of minimum degree of a vertex is valid for many graphs, including all bipartite graphs, and conjecture this bound ...

Let A be a unital simple C∗-algebra with tracial rank zero and X be a compact metric space. Suppose that h1, h2 : C(X) → A are two unital monomorphisms. We show that h1 and h2 are approximately unitarily equivalent if and only if [h1] = [h2] in KL(C(X), A) and τ ◦ h1(f) = τ ◦ h2(f) for every f ∈ C(X) and every trace τ of A.Adopting a theorem of Tomiyama, we introduce a notion of approximate con...

We introduce the tracial Rokhlin property for automorphisms of stably finite simple unital C*-algebras containing enough projections. This property is formally weaker than the various Rokhlin properties considered by Herman and Ocneanu, Kishimoto, and Izumi. Our main results are as follows. Let A be a stably finite simple unital C*-algebra, and let α be an automorphism of A which has the tracia...

We introduce a notion of covering dimension for Cuntz semigroups C⁎-algebras. This is always bounded by the nuclear C⁎-algebra, and subhomogeneous C⁎-algebras both dimensions agree. Z-stable have at most one. Further, semigroup simple, C⁎-algebra zero-dimensional if only has real rank zero or stably projectionless.

We show that, if A is a separable simple unital C-algebra which absorbs the Jiang–Su algebra Z tensorially and which has real rank zero and finite decomposition rank, then A is tracially AF in the sense of Lin, without any restriction on the tracial state space. As a consequence, the Elliott conjecture is true for the class of C-algebras as above which, additionally, satisfy the Universal Coeff...

Abstract. 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. Let mr(G) denote the minimum rank of all matrices in S(G), and mr+(G) the minimum rank of all positive semidefinite matrices in S(G). All graphs G with mr(G) = 2 and mr+(G) = k are chara...

The minimum rank of a sign pattern matrix is defined to be the smallest possible rank over all real matrices having the given sign pattern. The maximum nullity of a sign pattern is the largest possible nullity over the same set of matrices, and is equal to the number of columns minus the minimum rank of the sign pattern. Definitions of various graph parameters that have been used to bound maxim...