Algebraic connectivity and the characteristic set of a graph

Let Gbe a connected weighted graph on vertices {1,2,…,n} and L be the Laplacian matrix of GLet μ be the second smallest eigenvalue of L and Y be an eigenvector corresponding to μ. A characteristic vertex is a vertex v such that Y(v) = 0 and Y(w) ≠ 0 for some vertex w adjacent to v. An edge e with end vertices v,w is called a characteristic edge of G if Y(w) Y(v) < 0. The characteristic vertices and the characteristic edges together form the characteristic set of G. We investigate the characteristic set of an arbitrary graph. The relation between the characteristic set and nonnegative matrix theory is exploited. Bounds are obtained on the cardinality of the characteristic set. It is shown that if G is a connected graph with n vertices and m edges then the characteristic set has at most m − n + 2 elements. We use the description of the Moore-Penrose inverse of the vertex-edge incidence matrix of a tree to derive a classical result of Fiedler for a tree. Furthermore, an analogous result is obtained for an ei...