Families of graphs with maximum nullity equal to zero forcing number

Abstract

Abstract:The maximum nullity of a simple graph G, denoted M(G), is the largest possible nullity over all symmetric real matrices whose ijth entry is nonzero exactly when {i, j} is an edge in G for i ≠ j, and the iith entry is any real number. The zero forcing number of a simple graph G, denoted Z(G), is the minimum number of blue vertices needed to force all vertices of the graph blue by applying the color change rule. This research is motivated by the longstanding question of characterizing graphs G for which M(G) = Z(G). The following conjecture was proposed at the 2017 AIM workshop Zero forcing and its applications: If G is a bipartitesemiregular graph, then M(G) = Z(G). A counterexample was found by J. C.-H. Lin but questions remained as to which bipartite 3-semiregular graphs have M(G) = Z(G). We use various tools to nd bipartite families of graphs with regularity properties for which the maximum nullity is equal to the zero forcing number; most are bipartite 3-semiregular. In particular, we use the techniques of twinning and vertex sums to form new families of graphs for which M(G) = Z(G) and we additionally establish M(G) = Z(G) for certain Generalized Petersen graphs.

Maximum nullity and zero forcing of circulant graphs

“…Recent work [3] describes families of graphs for which equality holds in Theorem 1.1, that is, families of graphs G with M(G) = Z(G). If we let mr(G) = min{rank(A) | A ∈ S(G)}, then the rank theorem tells us that mr(G) + M(G) = n. Hence Theorem 1.1 demonstrates that zero forcing can provide a lower bound on the minimum rank of any symmetric matrix associated with a graph.…”

Computing the zero forcing number for generalized Petersen graphs

“…Hence, the vertices v Remark 2.4. In [2] the authors proved that Z(P (15r, 2)) = 6 and Z(P (24r, 5)) = 12 for all r ≥ 1. Also they proved Theorem 2.2 another way.…”

“…Therefore, the vertex v m+1 is the only white neighbor of the black vertex v m+1−k and u m+1 is the only white neighbor of the black vertex u m . If n = rk + s, then Z(P (n, k)) ≤ min{r(s + 2), 2k + 2}, where 1 ≤ s ≤ k − Remark In[2] the authors proved that Z(P (15r, 2)) = 6 and Z(P (24r, 5)) = 12 for all r ≥ 1. Also they proved Theorem 2.2 another way.…”

Techniques for determining equality of the maximum nullity and the zero forcing number of a graph

“…, n − 1 , for n ≥ 3 and k a positive integer less than n 2 (see [22]). In [4], the adjacency matrix was used to show that the maximum nullity is equal to the zero forcing number for certain generalized Petersen graphs. and the maximum nullity is attained by the adjacency matrix.…”

“…For simplicity, we consider the extended cube graphs with t ≤ k. The graph ECG(1, 1) is called the Bidiakis cube (see [4]). It was shown in [4,Proposition 5.1] that the maximum nullity and zero forcing number of the Bidiakis cube are the same, motivating the creation of the extended cube graphs. Observe that in ECG(t, k), as we draw it, the top endpoints of the vertical edges are 0, .…”