Note on positive semidefinite maximum nullity and positive semidefinite zero forcing number of partial 2-trees

Abstract

The maximum positive semideflnite nullity of a multigraph G is the largest possible nullity over all real positive semideflnite 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 semideflnite zero forcing number for simple graphs is extended to multigraphs; as for simple graphs, this parameter bounds the maximum positive semideflnite 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 semideflnite rank of outerplanar graphs and the tree cover number. Electron. J. Linear Algebra, 22:10-21, 2011.] is extended to show that Z +(G) = M +(G) = T(G) for a multigraph G of tree-width at most 2.