On trees with exactly one characteristic element

Let T be a tree and L be its Laplacian matrix. With respect to a given vector Y which gives a valuation of vertices of T, a vertex u of T is called a characteristic vertex if Y[u] = 0 and if there is a vertex w adjacent to u with Y[w] ≠ 0; and an edge {u, w} of T is called a characteristic edge if Y[u]Y[w] < 0. The characteristic set of T with respect to Y, denoted by C(T,Y), is defined as the collection of all characteristic vertices and characteristic edges of T. For the kth smallest eigenvalue λk (k ⩾ 2) of T, if λk > λk−1, the corresponding eigenvector Y of λk is called a k-vector. A tree T is called k-simple if |C(T,Y)|=1 for all k-vectors Y. We show that k-simple trees exist and characterize them. We also show the characteristic sets determined by all the k-vectors is the same, which is consistent with the property of 2-simple tree (i.e., arbitrary tree). Finally, we give some properties of the eigenvalues and eigenvectors of a k-simple tree.