Maximum nullity, zero forcing number and propagation time of ℓ-path graphs

Let G be a graph with each vertex is colored either white or black. A white vertex is changed to a black vertex when it is the only white neighbor of a black vertex (color-change rule). A zero forcing set S of a graph G is a subset of vertices G with black vertices, all other vertices G are white, such that after finitely many applications of the color-change rule all of vertices G becomes black. The zero forcing number of G is the minimum cardinality of a zero forcing set in G, denoted by Z(G). In this paper, we define ℓ−Path graphs. We give some ℓ−Path and ℓ−Ciclo graphs such that their maximum nullity are equal to their zero forcing number. Also, we obtain minimum propagation time and maximum propagation time for them.