Lower bounds for minimum semidefinite rank from orthogonal removal and chordal supergraphs

A new lower bound for the positive semidefinite minimum rank of a graph

“…The fact that ML 8 / ∈ E and E is closed under vertex sums and graph joins follows from the work in [14]. The motivation for the word "reduction" comes from a new interpretation of the OS-number developed by Mitchell et al [15] and described in Section 3 of this paper. The motivation for the word "signed" follows from the use of sign patterns to construct R s (G).…”

“…This was introduced by Booth et al [4], but we will follow the slightly different but equivalent formulation in Mitchell et al [15]. Consider a vector representation V ⊂ K d of a graph G = (V, E).…”

“…In this section we describe an interpretation of the OS-number which was introduced in [15]. Observe that Proposition 2.1 implies a lower bound for the semidefinite minimum rank: for example if v 1 is nondegenerate in G then…”

“…Originally the positive semidefinite zero forcing number and OS-number were only defined for simple graphs. However, Ekstrand et al [6] extended the definition of Z + (G) to multigraphs and Mitchell et al [15] extended the definition of OS(G) to multigraphs.…”

Colin de Verdiere parameters of chordal graphs

“…Given three vertices u, v, w, let e(u, v, w) be the product of the number of edges between u and v and the number of edges between w and v. Define two operations as follows: For G ⊖ v, in the induced subgraph G − v of G, between any u, w ∈ N (v), insert e(u, v, w) edges [21].…”