Locating Eigenvalues of Perturbed Laplacian Matrices of Trees

Abstract

ABSTRACT. We give a linear time algorithm to compute the number of eigenvalues of any perturbed Laplacian matrix of a tree in a given real interval. The algorithm can be applied to weighted or unweighted trees. Using our method we characterize the trees that have up to 5 distinct eigenvalues with respect to a family of perturbed Laplacian matrices that includes the adjacency and normalized Laplacian matrices as special cases, among others.

Locating Eigenvalues of a Symmetric Matrix whose Graph is Unicyclic

“…Algorithm 2 DiagonalizeW(T, α) [3] Input: underlying tree T = T (M) with ordered vertices v 1 , ..., v n and scalar α…”

“…unicyclic graphs where all the eigenvalues of their adjacency matrices are integers. Also in 2017, Braga and Rodrigues [3] developed an algorithm to locate eigenvalues of any perturbed Laplacian matrix of an edge-weighted tree, called DiagonalizeW. Given a real diagonal matrix D, the perturbed Laplacian matrix of a simple graph G with respect to D is the matrix L D (G) = D − A, where A = (a i j ) is the adjacency matrix of G. The adjacency, Laplacian, normalized Laplacian and signless Laplacian matrices, among others, are examples of perturbed Laplacian matrices.…”

Locating Eigenvalues of Symmetric Matrices - A Survey

“…As it turns out, this is always possible when the underlying graph associated with the original matrix is a tree, which has led to the seminal algorithm in this line of research due to Jacobs and Trevisan [37]. Their algorithm was designed specifically as an eigenvalue location algorithm for the eigenvalues of the adjacency matrix of trees (and was based on an earlier algorithm to compute the characteristic polynomial of such matrices [35]), but, as new applications were considered, the approach was naturally extended to other contexts, see [27] for the Laplacian matrix and [16] for general symmetric matrices whose underlying graph is a tree.…”