Failed zero forcing numbers of grassmann graphs

For a graph G with vertices colored either black or white, consider the following rule to change the colors: the color of a vertex which is the only white neighbor of a black vertex, changes from white to black. A proper subset S of the vertex set of G is called a failed zero forcing set if, regardless of how many times this rule is applied to a graph with the initial black vertices S, at least one white vertex always remains. The maximum size of such a subset is called the failed zero forcing number of G and is denoted by F(G). In this paper, we look at the failed zero forcing numbers of Grassmann graphs J_q(n, ۲) and prove that F(J_q(n, ۲))={n \brack ۲}_q - b'_۲(q), for n\geq ۵, where b'_۲(q) is the maximum number of points in the affine or projective plane of order q such that there is no line that passes through exactly one of these points. Moreover, using maximum arcs in the projective planes, we show that if q is a power of two, then F(J_q(n, ۲))={n \brack ۲}_q - (q + ۲).

کلیدواژه ها:

Failed zero forcing number ، Grassmann graph ، Projective plane ، Affine plane