Minimum Number Of Distinct Eigenvalues Research Articles

Bordering of symmetric matrices and an application to the minimum number of distinct eigenvalues for the join of graphs

An important facet of the inverse eigenvalue problem for graphs is to determine the minimum number of distinct eigenvalues of a particular graph. We resolve this question for the join of a connected graph with a path. We then focus on bordering a matrix and attempt to control the change in the number of distinct eigenvalues induced by this operation. By applying bordering techniques to the join of graphs, we obtain numerous results on the nature of the minimum number of distinct eigenvalues as vertices are joined to a fixed graph.

Diminimal families of arbitrary diameter

Given a tree T, let q(T) be the minimum number of distinct eigenvalues in a symmetric matrix whose underlying graph is T. It is well known that q(T)≥d(T)+1, where d(T) is the diameter of T, and a tree T is said to be diminimal if q(T)=d(T)+1. In this paper, we present families of diminimal trees of any fixed diameter. Our proof is constructive, allowing us to compute, for any diminimal tree T of diameter d in these families, a symmetric matrix M with underlying graph T whose spectrum has exactly d+1 distinct eigenvalues.

Sparsity of graphs that allow two distinct eigenvalues

The parameter q(G) of a graph G is the minimum number of distinct eigenvalues over the family of symmetric matrices described by G. It is shown that the minimum number of edges necessary for a connected graph G to have q(G)=2 is 2n−4 if n is even, and 2n−3 if n is odd. In addition, a characterization of graphs for which equality is achieved in either case is given.

Inverse eigenvalue and related problems for hollow matrices described by graphs

A hollow matrix described by a graph $G$ is a real symmetric matrix having all diagonal entries equal to zero and with the off-diagonal entries governed by the adjacencies in $G$. For a given graph $G$, the determination of all possible spectra of matrices associated with $G$ is the hollow inverse eigenvalue problem for $G$. Solutions to the hollow inverse eigenvalue problems for paths and complete bipartite graphs are presented. Results for related subproblems such as possible ordered multiplicity lists, maximum multiplicity of an eigenvalue, and minimum number of distinct eigenvalues are presented for additional families of graphs.

Minimum number of distinct eigenvalues allowed by a sign pattern

This article initiates the study of the minimum number of distinct eigenvalues allowed by a sign pattern. Non-trivial examples of sign patterns that allow matrices with only one eigenvalue include the potentially nilpotent and the spectrally arbitrary sign patterns, in contrast to the inverse eigenvalue problem for graphs. Necessary digraph cycle conditions are developed for a sign pattern to have a matrix realization with exactly one eigenvalue. Certain manipulations of sign patterns provide new n×n patterns that preserve the minimum number of eigenvalues allowed, including some Jacobian methods. The 2×2 and the irreducible 3×3 sign patterns are classified according to the minimum number of eigenvalues allowed by the pattern.

Orthogonal symmetric matrices and joins of graphs

We introduce a notion of compatibility for multiplicity matrices. This gives rise to a necessary condition for the join of two (possibly disconnected) graphs G and H to be the pattern of an orthogonal symmetric matrix, or equivalently, for the minimum number of distinct eigenvalues q of G∨H to be equal to two. Under additional hypotheses, we show that this necessary condition is also sufficient. As an application, we prove that q(G∨H) is either two or three when G and H are unions of complete graphs, and we characterise when each case occurs.

On the minimum number of distinct eigenvalues of a threshold graph

For a graph G, we associate a family of real symmetric matrices, S(G), where for any A∈S(G), the location of the nonzero off-diagonal entries of A are governed by the adjacency structure of G. Let q(G) be the minimum number of distinct eigenvalues over all matrices in S(G). In this work, we give a characterization of all connected threshold graphs G with q(G)=2. Moreover, we study the values of q(G) for connected threshold graphs with trace 2, 3, n−2, n−3, where n is the order of threshold graph. The values of q(G) are determined for all connected threshold graphs with 7 and 8 vertices with two exceptions. Finally, a sharp upper bound for q(G) over all connected threshold graph G is given.

The inverse eigenvalue problem for linear trees

We prove the sufficiency of the Linear Superposition Principle for linear trees, which characterizes the spectra achievable by a real symmetric matrix whose underlying graph is a linear tree. The necessity was previously proven in [10]. This is the most general class of trees for which the inverse eigenvalue problem has been solved. We explore many consequences, including the Degree Conjecture for possible spectra, upper bounds for the minimum number of eigenvalues of multiplicity 1, and the equality of the diameter of a linear tree and its minimum number of distinct eigenvalues.

An explicit upper bound on disparity for trees of a given diameter

ABSTRACT It is known that the minimum number of distinct eigenvalues c(T) of a symmetric matrix whose graph is a given tree T is at least the diameter d(T) of that tree. However, the disparity c(T) − d(T) can be positive. Using branch duplication and rooted seeds, the notion of the ‘most complex seed’ is introduced, and an explicit upper bound on the disparity is given for any tree of a given diameter.

On the Minimum Number of Distinct Eigenvalues in the Problem for a Tree Formed by Stieltjes Strings

We consider spectral problems related to small vibrations of a tree formed by Stieltjes strings. It is shown that the minimum number of distinct eigenvalues of this problem is equal to the maximum length (measured as the number of point masses) of paths in this tree.

The inverse eigenvalue problem of a graph: Multiplicities and minors

The inverse eigenvalue problem of a given graph G is to determine all possible spectra of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in G. Barrett et al. introduced the Strong Spectral Property (SSP) and the Strong Multiplicity Property (SMP) in [Generalizations of the Strong Arnold Property and the minimum number of distinct eigenvalues of a graph. Electron. J. Combin., 2017]. In that paper it was shown that if a graph has a matrix with the SSP (or the SMP) then a supergraph has a matrix with the same spectrum (or ordered multiplicity list) augmented with simple eigenvalues if necessary, that is, subgraph monotonicity. In this paper we extend this to a form of minor monotonicity, with restrictions on where the new eigenvalues appear. These ideas are applied to solve the inverse eigenvalue problem for all graphs of order five, and to characterize forbidden minors of graphs having at most one multiple eigenvalue.

On Orthogonal Matrices with Zero Diagonal

This paper considers real orthogonal $n\times n$ matrices whose diagonal entries are zero and off-diagonal entries nonzero, which are referred to as $\OMZD(n)$. It is shown that there exists an $\OMZD(n)$ if and only if $n\neq 1,\ 3$, and that a symmetric $\OMZD(n)$ exists if and only if $n$ is even and $n\neq 4$. Also, a construction of $\OMZD(n)$ obtained from doubly regular tournaments is given. Finally, the results are applied to determine the minimum number of distinct eigenvalues of matrices associated with some families of graphs, and the related notion of orthogonal matrices with partially-zero diagonal is considered.

Diagonalizable matrices whose graph is a tree: the minimum number of distinct eigenvalues and the feasibility of eigenvalue assignments

Abstract Considered are combinatorially symmetric matrices, whose graph is a given tree, in view of the fact recent analysis shows that the geometric multiplicity theory for the eigenvalues of such matrices closely parallels that for real symmetric (and complex Hermitian) matrices. In contrast to the real symmetric case, it is shown that (a) the smallest example (13 vertices) of a tree and multiplicity list (3, 3, 3, 1, 1, 1, 1) meeting standard necessary conditions that has no real symmetric realizations does have a diagonalizable realization and for arbitrary prescribed (real and multiple) eigenvalues, and (b) that all trees with diameter < 8 are geometrically di-minimal (i.e., have diagonalizable realizations with as few of distinct eigenvalues as the diameter). This re-raises natural questions about multiplicity lists that proved subtly false in the real symmetric case. What is their status in the geometric multiplicity list case?

A Nordhaus–Gaddum conjecture for the minimum number of distinct eigenvalues of a graph

We propose a Nordhaus–Gaddum conjecture for q(G), the minimum number of distinct eigenvalues of a symmetric matrix corresponding to a graph G: for every graph G excluding four exceptions, we conjecture that q(G)+q(Gc)≤|G|+2, where Gc is the complement of G. We compute q(Gc) for all trees and all graphs G with q(G)=|G|−1, and hence we verify the conjecture for trees, unicyclic graphs, graphs with q(G)≤4, and for graphs with |G|≤7.

Geometric Parter–Wiener, etc. theory

For real symmetric, or complex Hermitian, matrices whose graph is a tree, there is a well-developed (via several papers) theory about the possible multiplicities of the eigenvalues. It includes a theory of vertices whose removal increases a multiplicity (the “Parter–Wiener, etc. theory” and the “downer branch mechanism”), how to determine maximum multiplicity, lower bounds for the minimum number of distinct eigenvalues, etc. Remarkably, a great deal of this theory may be generalized to geometric multiplicities of general matrices over a field (with very different proofs). We show here what parts of this theory generalize, and, in the process, review the theory.

Generalizations of the Strong Arnold Property and the Minimum Number of Distinct Eigenvalues of a Graph

For a given graph $G$ and an associated class of real symmetric matrices whose diagonal entries are governed by the adjacencies in $G$, the collection of all possible spectra for such matrices is considered. Building on the pioneering work of Colin de Verdière in connection with the Strong Arnold Property, two extensions are devised that target a better understanding of all possible spectra and their associated multiplicities. These new properties are referred to as the Strong Spectral Property and the Strong Multiplicity Property. Finally, these ideas are applied to the minimum number of distinct eigenvalues associated with $G$, denoted by $q(G)$. The graphs for which $q(G)$ is at least the number of vertices of $G$ less one are characterized.

Diameter minimal trees

Using the method of seeds and branch duplication, it is shown that for every tree of diameter , there is an Hermitian matrix with as few as distinct eigenvalues (a known lower bound). For diameter 7, some trees require 8 distinct eigenvalues, but no more; the seeds for which 7 and 8 are the worst case are classified. For trees of diameter , it is shown that, in general, the minimum number of distinct eigenvalues is bounded by a function of . Many trees of high diameter permit as few of distinct eigenvalues as the diameter and a conjecture is made that all linear trees are of this type. Several other specific, related observations are made.

Minimum number of distinct eigenvalues of graphs

The minimum number of distinct eigenvalues, taken over all real symmetric matrices compatible with a given graph G, is denoted by q(G). Using other parameters related to G, bounds for q(G) are proven and then applied to deduce further properties of q(G). It is shown that there is a great number of graphs G for which q(G) = 2. For some families of graphs, such as the join of a graph with itself, complete bipartite graphs, and cycles, this minimum value is obtained. Moreover, examples of graphs G are provided to show that adding and deleting edges or vertices can dramatically change the value of q(G). Finally, the set of graphs G with q(G) near the number of vertices is shown to be a subset of known families of graphs with small maximum multiplicity.

Unordered multiplicity lists of a class of binary trees

Associated with a given graph G on n vertices 1,2,…,n, is the set S(G) of all n×n real symmetric matrices A=[aij] whose off-diagonal entries are placed according to the edges of G, i.e., for i≠j, aij≠0 if and only if vertices i and j are adjacent. In this paper we study spectral properties of matrices in S(G) for a class of binary trees G. We first show that a matrix A in S(G) has no eigenvalues of multiplicity 4 or more, at most one eigenvalue of multiplicity 3, and at least three simple eigenvalues. We then completely determine the unordered multiplicity lists of these binary trees. As a consequence, it is shown that the minimum number of distinct eigenvalues of a matrix in S(G) is one more than the diameter of G.

Branch duplication for the construction of multiple eigenvalues in an Hermitian matrix whose graph is a tree

Suppose that the eigenvalues of an Hermitian matrix A whose graph is a tree T are known, as well as the eigenvalues of the principal submatrix of A corresponding to a certain branch of T. A method for constructing a larger tree T ', in which the branch is ‘`duplicated’', and an Hermitian matrix A′ whose graph is T ' is described. The eigenvalues of A' are all of those of A, together with those corresponding to the branch, including multiplicities. This idea is applied (1) to give a solution to the inverse eigenvalue problem for stars, (2) to prove that the known diameter lower bound, for the minimum number of distinct eigenvalues among Hermitian matrices with a given graph, is best possible for trees of bounded diameter, and (3) to increase the list of trees for which all possible lists for the possible spectra are know. A generalization of the basic branch duplication method is presented.