Minimum rank and maximum eigenvalue multiplicity of symmetric tree sign patterns

On the minimum semidefinite rank of a simple graph

“…The real symmetric problem, as well as the corresponding one for matrices in H(G), have since been considered by many others [1,2,5,6,8]. In this article, we consider the minimum semidefinite rank (msr) problem, which seeks to determine the msr of a graph G, msr(G), the minimum rank among matrices in P(G).…”

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

“…With slightly different notation, mr(G, f ) has been computed when G is a tree [5,12]. Several authors have also considered a non-symmetric variant of this parameter: given a m-by-n sign pattern Z, what is the minimum rank of a matrix A with sign pattern Z?…”

On the minimum rank of not necessarily symmetric matrices: A preliminary study

“…The path cover number P (T ) is the minimum number of vertex disjoint paths that cover all the vertices of T . In [5] a generalization of ∆ was used because the obvious extension of the definition of path cover number, namely the minimum number of vertex disjoint paths that cover all the vertices of T , need not be equal to maximum nullity for a loop tree T . Here we introduce a different definition of path cover number, which coincides with that in [9] for trees, and show in Section 5 that using our Definition 4.19, path cover number, maximum nullity, and another readily computable parameter, the zero forcing number, are equal for any ditree.…”

“…In [9], for a simple tree T the parameter ∆(T ) was defined to be the maximum of p − q such that there is a set of q vertices whose deletion leaves p paths, and it was shown that M(T ) = ∆(T ) = P (T ). In [5] the definition of ∆ was extended to the parameter C 0 (T ) defined for a loop tree T , and it was shown that C 0 (T ) = M(T ). In [10] Mikkelson extended the applicability of this result by showing that M F (T ) = M(T ) for every field F of order greater than two.…”

“…[5]). For convenience and completeness, we reproduce and translate the necessary terminology and the algorithm into the language of ditrees and minimum rank (in [5] it is stated more generally to include sign patterns and nonzero eigenvalues).…”