Maximum nullity and zero forcing number on graphs with maximum degree at most three

Abstract

A dynamic coloring of a graph G starts with an initial subset F⊆V(G) of colored vertices, while all the remaining vertices are non-colored. At each time step, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set F is called a zero forcing set of G if, by iteratively applying the forcing process, every vertex in G becomes colored. The zero forcing number of G, denoted by F(G), is the cardinality of a minimum zero forcing set of G. The maximum nullity of G, denoted by M(G), is the largest possible nullity over all |V(G)| by |V(G)| real symmetric matrices A whose non-diagonal entries are non-zero if the corresponding vertices are adjacent in G and with no restriction for its diagonal entries. In this paper, we characterize all graphs G of order n, maximum degree at most three, and F(G)=3. Also we classify these graphs with their maximum nullity.