Friday, October 28, 2011
Hall 1-2 (San Jose Convention Center)
A graph is denoted G=(V,E), where V is a nonempty set of vertices, E is a set of edges, and each edge is two-element subset of the set of vertices V. Our research focuses on simple undirected graphs and a type of graph parameter called the zero forcing number. A specific type of zero forcing number, positive semidefinite zero forcing number, was introduced by Barioli, et al. in 2010. A positive semidefinite zero forcing set for a graph G is a subset B of V, such that when B is initially colored black, all vertices of G are colored black when the positive semidefinite color change rule is carried out to completion. We call the minimum cardinality over all such B the positive semidefinite zero forcing number of G, and denote this quantity Z+(G). A positive semidefinite matrix representation of a graph G on n vertices is a n-square positive semidefinite matrix A=[aij], where aij is nonzero if the vertices i and j of G are adjacent and zero otherwise. Two interesting parameters based on the family of positive semidefinite matrices representing a graph G are positive semidefinite minimum rank and positive semidefinite maximum nullity of G, denoted by mr+(G) and M+(G). It is well known that M+(G) ≤ Z+(G) for all graphs. We investigate the effect of deleting one vertex of G, along with its adjacent edges, on the parameters mr+(G), M+(G), and Z+(G).