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 vertexis changed to a black vertex when it is the only white neighbor of a black vertex (color-changerule). a zero forcing set s of a graph g is a subset of vertices g with black vertices, all othervertices g are white, such that after finitely many applications of the color-change rule allof vertices g becomes black. the zero forcing number of g is the minimum cardinality ofa zero forcing set in g, denoted by z(g). in this paper, we define ℓ−path graphs. we givesome ℓ−path and ℓ−ciclo graphs such that their maximum nullity are equal to their zeroforcing number. also, we obtain minimum propagation time and maximum propagation timefor them.