Eigenspaces of Preserved Graph Laplacian Eigenvalues under Positive Edge-Weight Addition

Abstract

We study how the eigenspace associated with a fixed graph Laplacian eigenvalue changes under positive edge updates, allowing both new edges and increases in existing edge weights. We show that a preserved eigenvalue may have inherited eigenvectors from the original eigenspace and emergent eigenvectors outside it. We characterize the inherited eigenspace and show that it is independent of the update weights. We characterize its orthogonal complement within the preserved eigenspace through a symmetric matrix whose size is at most the number of updated edges, yielding a multiplicity formula and an explicit reconstruction of the updated eigenspace. For fixed updated edges, we characterize when emergence is possible and show that, whenever it is, equal update weights suffice. The exceptional weights form the zero set of a nonzero multi-affine polynomial in reciprocal variables and have Lebesgue measure zero. Finally, emergence cannot occur at either of the two smallest Laplacian eigenvalues, counting multiplicities.