Unordered multiplicity lists of wide double paths

Abstract

Recently, Kim and Shader analyzed the multiplicities of the eigenvalues of a Φ-binary tree. We carry this discussion forward extending their results to a larger family of trees, namely, the wide double path, a tree consisting of two paths that are joined by another path. Some introductory considerations for dumbbell graphs are mentioned regarding the maximum multiplicity of the eigenvalues. Lastly, three research problems are formulated.

The maximum of the minimal multiplicity of eigenvalues of symmetric matrices whose pattern is constrained by a graph

“…, where a 1 > a 2 − t and t > 0, then the eigenvalues of matrix D 0 , defined in (6), are equal to a 1 +t, a 1 +t, a 2 −t, a 2 −t and the orthogonal matrix that diagonalizes D 0 is equal to…”

“…. , λ n } that can be the spectrum of a matrix A ∈ S(G), and is known as the Inverse eigenvalue problem for G. This question and the related question of characterizing all possible multiplicities of eigenvalues of matrices in S(G) have been studied primarily for trees [6,11,13,14]. A subproblem to the inverse eigenvalue problem for graphs that has attracted a lot of attention over the years is that of minimizing the rank of all A ∈ S(G).…”

Graphs that allow all the eigenvalue multiplicities to be even

“…, λ n } that can be the spectrum of a matrix A ∈ S(G) is known as the Inverse eigenvalue problem for G, and it is hard to answer in general. This question and the related question of characterizing all possible multiplicities of eigenvalues of matrices in S(G) has been studied primarily for trees [6,9,10,11]. The subproblem to the inverse eigenvalue problem for graphs that has attracted a lot of attention over the recent years is that is equivalent to finding the maximal multiplicity of an eigenvalue of A ∈ S(G).…”

Inverse eigenvalue problems for acyclic matrices whose graph is a dense centipede

“…In particular, we thank the reviewers for bringing the paper [5] on multiplicities of eigenvalues to our notice. We hope that this paper will be helpful in studying the multiplicities of eigenvalues for the family of dense centipedes.…”