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

The maximum nullity M(G) and the Colin de Verdiere type parameter xi(G) both consider the largest possible nullity over matrices in 8(G), which is the family of real symmetric matrices whose i, j-entry, i not equal j, is nonzero if i is adjacent to j, and zero otherwise; however, (G) restricts to those matrices A in 8(G) with the Strong Arnold Property, which means X = O is the only symmetric matrix that satisfies A circle X = O, I circle X = O, and AX = O. This paper introduces zero forcing parameters Z(SAP)(G) and Z(vc)(G), and proves that Z(SAP) (G) = O implies every matrix A is an element of S(G) has the Strong Arnold Property and that the inequality M(G) - Z(vc)(G) <= xi(G) holds for every graph G. Finally, the values of xi(G) are computed for all graphs up to 7 vertices, establishing xi(G) = left perpendicularZright perpendicular(G) for these graphs. (C) 2016 Elsevier Inc. All rights reserved.