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

Abstract

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.

Lower bounds for positive semidefinite zero forcing and their applications

“…Since every series-parallel graph is a partial 2-tree, we have Z + (G i ) = tc(G i ) Ekstrand et al (2012). So, from Lemma 6.3 and Theorem 6.6, we have…”

“…Note that Ekstrand et al (2012) showed that Z + (G) = tc(G) = M + (G) for any graph that is a partial 2-tree. Since every series-parallel graph is a partial 2-tree, we have the following.…”

“…Ekstrand et al (2012) showed that Z + (G) = tc(G) = M + (G) for any graph that is a partial 2-tree. Since each cactus is a partial 2-tree, we have Z + (G) = tc(G) = M + (G) for any cactus G. The following corollary enhances this result by establishing a relation between Z + (G) and the number of simple cycles in G. Proof Let G = (V, E) be a cactus with k blocks.…”

“…The most widely-studied variant of zero forcing, called the positive semidefinite zero forcing (PSZF for short), was introduced in Barioli et al (2010) [see also Ekstrand et al (2012Ekstrand et al ( , 2013; Fallat et al (2015) for recent results]. Similar to the zero forcing number, a primary reason to study the PSZF number of a graph is its relationship to the maximum positive semidefinite nullity of certain positive semidefinite symmetric matrices associated with the graph.…”

Path cover number, maximum nullity, and zero forcing number of oriented graphs and other simple digraphs

“…With this notation, the GCC can be denoted GCC M . The graph complement conjecture for zero forcing number, Z(G) + Z(G) ≥ |G| − 2, denoted GCC Z , is actually the graph complement theorem for zero forcing [Ekstrand et al 2012]. However, as the following example shows, the GCC Z does not hold for digraphs, and since for any digraph M( ) ≤ Z( ), the GCC M does not hold for digraphs.…”