Abstract. Let G = (V, E) be a multigraph with no loops on the vertex set V = {1, 2, . . . , n}. Define S + (G) as the set of symmetric positive semidefinite matrices A = [a ij ] with a ij = 0, i = j, if ij ∈ E(G) is a single edge and a ij = 0, i = j, if ij / ∈ E(G). Let M + (G) denote the maximum multiplicity of zero as an eigenvalue of A ∈ S + (G) and mr + (G) = |G| − M + (G) denote the minimum semidefinite rank of G. The tree cover number of a multigraph G, denoted T (G), is the minimum number of vertex disjoint simple trees occurring as induced subgraphs of G that cover all of the vertices of G.
The minimum semidefinite rank (msr) of a graph is defined to be the minimum rank among all positive semidefinite matrices whose zero/ nonzero pattern corresponds to that graph. We recall some known facts and present new results, including results concerning the effects of vertex or edge removal from a graph on msr.
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