2023/01/26 by Shaun Fallat, H. Tracy Hall, Fallat, Shaun M. +11
Chemistry · Computer Science · Mathematics · #05C50 #15A18 (Secondary) #15A29 (Primary) #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms #Molecular spectroscopy and chirality #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2301.11073
openalex publication_date 2023/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If G is a graph and m is an ordered multiplicity list which is realizable by at least one symmetric matrix with graph G, what can we say about the eigenvalues of all such realizing matrices for m? It has sometimes been tempting to expect, especially in the case that G is a tree, that any spacing of the multiple eigenvalues should be realizable. In 2004, however, F. Barioli and S. Fallat produced the first counterexample: a tree on 16 vertices and an ordered multiplicity list for which every realizing set of eigenvalues obeys a nontrivial linear constraint. We extend this by giving an infinite family of trees and ordered multiplicity lists whose sets of realizing eigenvalues are very highly constrained, with at most 5 degrees of freedom, regardless of the size of the tree in this family. In particular, we give the first examples of multiplicity lists for a tree which impose nontrivial nonlinear eigenvalue constraints and produce an ordered multiplicity list which is achieved by a unique set of eigenvalues, up to shifting and scaling.