Using a new zero forcing process to guarantee the Strong Arnold Property

The maximum nullity M(G)M(G) and the Colin de Verdière type parameter ξ(G)ξ(G) both consider the largest possible nullity over matrices in S(G)S(G), which is the family of real symmetric matrices whose i,ji,j-entry, i≠ji≠j, is nonzero if i is adjacent to j  , and zero otherwise; however, ξ(G)ξ(G) restricts to those matrices A   in S(G)S(G) with the Strong Arnold Property, which means X=OX=O is the only symmetric matrix that satisfies A∘X=OA∘X=O, I∘X=OI∘X=O, and AX=OAX=O. This paper introduces zero forcing parameters ZSAP(G)ZSAP(G) and Zvc(G)Zvc(G), and proves that ZSAP(G)=0ZSAP(G)=0 implies every matrix A∈S(G)A∈S(G) has the Strong Arnold Property and that the inequality M(G)−Zvc(G)≤ξ(G)M(G)−Zvc(G)≤ξ(G) holds for every graph G  . Finally, the values of ξ(G)ξ(G) are computed for all graphs up to 7 vertices, establishing ξ(G)=⌊Z⌋(G)ξ(G)=⌊Z⌋(G) for these graphs.