On the multiplicity of-1 as an eigenvalue of a tree with given number of pendant vertices

Let T be a tree with vertex set V(T) = {v(1), v(2), ... , v(n)}. The adjacency matrix A(T) of T is an n x n matrix (a(ij)), where a(ij) = a(ji) = 1 if v(i) is adjacent to v(j) and a(ij) = 0 if otherwise. In this paper, we consider the multiplicity of -1 as an eigenvalue of A(T), which is written as m(T,-1). It is proved that among all trees T with p >= 2 pendant vertices, the maximum value of m(T,-1) is p-1, and for a tree T with p >= 2 pendant vertices, m(T,-1) = p - 1 if and only if T = P-n with n = 2(mod 3), or T is a tree in which d(v, u) = 2(mod 3) for any pendant vertex v and any major vertex u of T, where a major vertex is a vertex of degree at least 3 and d(v, u) is the distance between v and u.