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The maximum positive semidefinite nullity of a multigraph G is the largest possible nullity over all real positive semidefinite matrices whose (i, j)th entry (for i = j) is zero if i and j are not adjacent in G, is nonzero if {i, j} is a single edge, and is any real number if {i, j} is a multiple edge. The definition of the positive semidefinite zero forcing number for simple graphs is extended to multigraphs; as for simple graphs, this parameter bounds the maximum positive semidefinite nullity from above. The tree cover number T(G) is the minimum number of vertex disjoint induced simple trees that cover all of the vertices of G. The result that M + (G) = T(G) for an outerplanar multigraph G [F. Barioli et al. Minimum semidefinite rank of outerplanar graphs and the tree cover number.

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Abstract. The problem of diagonalizing hermitian matrices of continuous fiunctions was studied by Grove and Pederson in 1984. While diagonalization is not possible in general, in the presence of differentiability conditions we are able to obtain positive results in the case of 2 × 2 matrices. It remains open whether our results can be extended to n × n matrices.