Let G be a connected graph with n vertices and let x=(x 1,..., x n) be a real vector. A positive (negative) sign graph of the vector x is a maximal connected subgraph of G on vertices x i>0 (x i<0). For an eigenvalue of a generalized Laplacian of a tree: We characterize the maximal number of sign graphs of an eigenvector. We give an O(n 2) time algorithm to find an eigenvector with maximum number of sign graphs and we show that finding an eigenvector with minimum number of sign graphs is an NP-complete problem. © 2002 Elsevier Science Inc. All rights reserved.
Biyikoǧlu, T. (2003). A discrete nodal domain theorem for trees. Linear Algebra and Its Applications, 360, 197–205. https://doi.org/10.1016/S0024-3795(02)00451-2