The characteristic set with respect to the k-maximal vectors of a tree
نویسندگان
چکیده
Let T be a tree on n vertices and L(T ) be its Laplacian matrix. The eigenvalues and eigenvectors of T are respectively referred to those of L(T ). With respect to a given eigenvector Y of T , a vertex u of T is called a characteristic vertex if Y [u] = 0 and there is a vertex w adjacent to u with Y [w] 6= 0; an edge e = (u, w) of T is called a characteristic edge if Y [u]Y [w] < 0. C(T, Y ) denotes the characteristic set of T with respect to the vector Y , which is defined as the collection of all characteristic vertices and characteristic edges of T corresponding to Y . Let λ1(T ) ≤ λ2(T ) ≤ · · · ≤ λn(T ) be the eigenvalues of a tree T on n vertices. An eigenvector is called a k-vector (k ≥ 2) of T if the eigenvalue λk(T ) associated by this eigenvector satisfies λk(T ) > λk−1(T ). The k-vector Y of T is called k-maximal if C(T, Y ) has maximum cardinality among all k-vectors of T . In this paper, the characteristic set with respect to any k-maximal vector of a tree is investigated by exploiting the relationship between the cardinality of the characteristic set and the structure of this tree. With respect to any k-maximal vector Y of a tree T , the structure of the trees T satisfying |C(T, Y )| = k − 1− t for any t (0 ≤ t ≤ k − 2) are characterized.
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