www.elsevier.com/locate/laa
The structure of matrices with a maximum multiplicity
eigenvalue
Charles R. Johnson
a, António Leal Duarte
b, Carlos M. Saiago
c,,∗,1 aDepartment of Mathematics, College of William and Mary, P.O. Box 8795, Williamsburg, VA 23187-8795, USAbCMUC, Department of Mathematics, University of Coimbra, Apartado 3008, 3001-454 Coimbra, Portugal cDepartamento de Matemática, Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa,
2829-516 Quinta da Torre, Portugal
Received 6 August 2007; accepted 11 April 2008
Submitted by S. Kirkland
Abstract
There is remarkable and distinctive structure among Hermitian matrices, whose graph is a given treeT
and that have an eigenvalue of multiplicity that is a maximum forT. Among such structure, we give several new results: (1) no vertex ofTmay be “neutral”; (2) neutral vertices may occur if the largest multiplicity is less than the maximum; (3) every Parter vertex has at least two downer branches; (4) removal of a Parter vertex changes the status of no other vertex; and (5) every set of Parter vertices forms a Parter set. Statements (3), (4) and (5) are also not generally true when the multiplicity is less than the maximum. Some of our results are used to give further insights into prior results, and both the review of necessary background and the development of new structural lemmas may be of independent interest.
© 2008 Elsevier Inc. All rights reserved.
Keywords: Hermitian matrices; Eigenvalues; Multiplicities; Maximum multiplicity; Trees; Path cover number; Parter vertices
∗Corresponding author.
E-mail addresses:[email protected](C.R. Johnson),[email protected](A. Leal Duarte),[email protected](C.M. Saiago).
1This work was done within the activities ofCentro de Matemática e Aplicaçõesda Universidade Nova de Lisboa. 0024-3795/$ - see front matter(2008 Elsevier Inc. All rights reserved.
1. Introduction
For a simple (labelled) graphG, letS(G)denote the set of all Hermitian matricesA=(aij) whose graph isG, i.e., fori /=j,aij =/ 0 if and only if{i, j}is an edge ofG. No restrictions is placed upon the diagonal entries ofA(other than reality) byG. Since all that we consider about Ais permutation similarity invariant, the labelling ofGis incidental (for reference only), and we generally considerGto be unlabelled.
We are interested inσ (A),A∈S(G), especially the multiplicities of the eigenvalues, and, particularly, the case in whichG=T, a tree. We use conventional submatrix, subgraph notation and often move informally between the two. Ifαis a subset of the indicesN = {1, . . . , n}of A(resp. vertices ofG),A(α)(resp.G−α) denotes the principal submatrix ofA(resp. induced subgraph ofG) resulting from deletion of the indices (resp. vertices)α.A({i})(resp.G− {i}) is abbreviated byA(i)(resp.G−i).A[α]orG[α]denotes the principal submatrix or induced subgraph resulting from keeping only the indices or vertices α. IfG′=G[α] we often write A[G′], meaning the principal submatrixA[α]. In caseT is a tree andv is a vertex of degreek, T −vis a forest consisting ofkbranches (trees) atT:T1, . . . , Tk. We often assume this branch notation without further explanation.
LetmA(λ)denote the multiplicity ofλas a root of the characteristic polynomial of then-by-n matrixA. We allowmA(λ)=0. Since we consider Hermitian matricesA, there is no ambiguity between algebraic and geometric multiplicity. The vertexvof a treeT is calledParterforλin A∈S(T )(Parter, for short) if
mA(v)(λ)=mA(λ)+1.
In the only two other possibilities (because of the interlacing inequalities [2])
mA(v)(λ)=mA(λ) and mA(v)(λ)=mA(λ)−1,
the vertexvis calledneutralor adowner, respectively. In general, each of these can easily occur and the theory is well developed in [6,9], with a summary of known results in [4]. A set of vertices αof cardinalitykis called aParter setif
mA(α)(λ)=mA(λ)+k.
In general a set of Parter vertices, each of which is Parter forλ,AandT isnota Parter set. (Examples will be discussed later.) By thestatusof a vertex, relative to a particularλ,A,T, we mean its classification as Parter, neutral or a downer.
Our purpose here is to more deeply understand the special structure occurring when an Hermi-tianA∈S(T )has an eigenvalueλsuch thatmA(λ)=M(T ), the maximum possible multiplicity for an eigenvalue among matrices inS(T ).
2. Necessary background and motivation
Here, we summarize some necessary background, organized into subsections for easy ref-erence. As elsewhere, we benignly blur the distinction between tree and matrix and between subgraph and submatrix. In all statements, there is an implicit identified eigenvalue and ma-trix relative to a given tree. For more thorough explanations or further background, see [6,4].
2.1. Maximum multiplicity, path covers and residual path maximizing sets
For a given tree T, the maximum possible multiplicityM(T )may be characterized in two combinatorial ways [3]. Apath coverof a tree is a collection of vertex disjoint, induced paths ofT that cover the vertices of the tree. The least number of paths in a path cover is thepath cover numberP (T ). A residual path maximizing(RPM) set for T is a collection of q ver-tices ofT, whose removal fromT leaves a forest of p paths in such a way that p−q is a maximum. The maximum multiplicityM(T )is equal to bothP (T )and this maximump−q. A matrixA∈S(T ) achievingmA(λ)=M(T )may be constructed by assigning λ as small-est eigenvalue to each submatrix of A corresponding to a path after removal of an RPM set for T. The vertices of this RPM set will then form a Parter set (see Section 2.3) for λ inA.
2.2. Parter vertices and downer branches
The survey [6], which contains original material, describes the relation between a Parter vertex vfor an eigenvalueλofA∈S(T )and neighbors ofv that are downer vertices forλin their branches. For each eigenvalueλsuch thatmA(λ)2, there exist Parter vertices (and they may exist for eigenvalues of multiplicity 0 or 1). Ifvis a Parter vertex (forλinA∈S(T )), then there is a neighborui ofvthat is a downer vertex forλin its branchTi, i.e., inA[Ti]. We call such a branchTiadowner branchfor the Parter vertexv, and, conversely, ifvhas a downer branch, then vis Parter forλ.
This downer branch mechanism for identifying Parter vertices is subtle and very important. In general, there may be only one downer branch at a Parter vertex, but we show that ifλattains maximum multiplicity, there must be at least two, a strong structural distinction in the maximum multiplicity case.
Wheneverλ∈σ (A(v))for any vertexvofT such thatA∈S(T ), there will be at least one Parter vertex forλ(not necessarilyv), even ifmA(λ)=0 ormA(λ)=1. In this event, the above downer branch mechanism is still in place.
2.3. Parter sets and fragmenting Parter sets
short) if its removal fromT leaves a forest in which the multiplicity ofλ is at most 1 in the submatrix of Aassociated with each tree of the forest. Of course, FPS’s always exist and any maximal (wrt inclusion) Parter set will be one. An FPS is a “fully fragmenting Parter set” if no vertex of any of the trees that remain is Parter, i.e., none of the trees may be further broken down.
In general, the trees that remain after an FPS or fully FPS is removed may be rather arbitrary, but, we will see that ifλattainsM(T ), removal of any Parter vertex leaves trees, in each of which maximum multiplicity is attained byλ. Thus, ifλattainsM(T ), removal of an FPS leaves only paths, and, in fact, any FPS is an RPM, another structural distinction.
2.4. Facts about paths
In all of this, facts about paths are especially basic and important. A rather complete theory was developed in [6].
It has long been known that ifT is a path,M(T )=1, i.e., the eigenvalues ofA∈S(T )are distinct. Moreover, paths are the only graphs for which maximum multiplicity is 1 [1], and this is obvious among trees. It is also known that removal of a pendent vertex from a path results in strict interlacing of eigenvalues, which need not be the case among trees that are not paths. A pendent vertex of a path is a downer vertex for any eigenvalue occurring in that path. Removal of an interior vertex from a path may not result in strict interlacing, but ifλappears in bothT andT −v, thenvis Parter forλandbothbranches atvare downer branches atv. Of course, ifλ appears inT (i.e., as an eigenvalue ofA∈S(T )), thenλattainsM(T )=1, and we will see that the occurrence of two downer branches generalizes to all cases of maximum multiplicity.
In the context of general trees, we say that a path ispendentat a vertexv, not in the path, if the only vertex of the path that is adjacent tovis a degree one vertex.
3. New structural lemmas
Lemma 1. LetT be a tree onnvertices, A∈S(T ),and suppose that there is an eigenvalueλ
ofAof multiplicityM(T ).Ifvis a Parter vertex forλwe have the following.
(1)The degree ofvinT is at least2.
(2)IfT1, . . . , Tkare the branches ofT atv,thenmA[Ti](λ)=M(Ti),i=1, . . . , k.
Proof. For (1), it suffices to note that any Parter vertex for an eigenvalue belongs to an FPS and, in case the multiplicity isM(T ), each FPS is an RPM set ofT. Since a pendent vertex cannot belong to an RPM set (because max[p−q]cannot be achieved when a pendent vertex is among theq removed vertices fromT in order to leavepcomponents) we conclude that a Parter vertex for an eigenvalue of multiplicityM(T )must have, at least, degree 2. This also follows from the fact thatP (T −v)P (T )whenvis pendant inT.
For (2), note that since A(v)=A[T1] ⊕ · · · ⊕A[Tk] and v is Parter for λ, we have that mA(v)(λ)=M(T )+1=mA[T1](λ)+ · · · +mA[Tk](λ). In order to obtain a contradiction, we
suppose thatmA[Ti](λ) < M(Ti)for a particular branch ofT atv. If we consider a matrixB∈
S(T ) such thatmB[T
i](λ)=M(Ti),i=1, . . . , k, we would have mB(v)(λ)=M(T1)+ · · · +
Lemma 2.LetT be a tree onnvertices, A∈S(T )andλbe an eigenvalue ofA.Letu1andu2
be two adjacent Parter vertices forλrelative toA.We denote byT1the component ofT −u2
containingu1and byT2the component ofT −u1containingu2.Then we have the following:
(1)u1is a Parter vertex forλrelative toA[T1]if and only ifu2is a Parter vertex forλrelative
toA[T2].
(2)u1is a downer vertex forλrelative toA[T1]if and only ifu2is a downer vertex forλrelative
toA[T2].
(3)Neitheru1is a neutral vertex forλrelative toA[T1]nor isu2a neutral vertex forλrelative
toA[T2].
Proof. First note thatA(u1)=A[T1−u1] ⊕A[T2]andA(u2)=A[T1] ⊕A[T2−u2]and, since u1andu2are Parter vertices forλrelative toA, we havemA(u1)(λ)=mA(u2)(λ)=mA(λ)+1.
If u1is Parter for λrelative toA, thenu2being Parter for λrelative toA[T2]implies that mA({u1,u2})(λ)=mA(λ)+2. Thus, the initial removal ofu2means thatu1is Parter forλrelative
toA[T1]. Similarly,u1being Parter forλrelative toA[T1]implies thatu2is Parter forλrelative toA[T2], which verifies (1). (Note that this argument does not depend upon the adjacency ofu1 andu2.)
For (2), we suppose without loss of generality thatu1is a downer vertex forλrelative toA[T1]. Since
mA[T1](λ)+mA[T2−u2](λ)=mA(u2)(λ)
=mA(u1)(λ)
=mA[T1−u1](λ)+mA[T2](λ)
=mA[T1](λ)−1+mA[T2](λ)
we havemA[T2−u2](λ)=mA[T2](λ)−1, i.e.,u2is a downer vertex forλrelative toA[T2].
For (3), in order to obtain a contradiction, we suppose thatu1is a neutral vertex forλrelative toA[T1], i.e.,mA[T1−u1](λ)=mA[T1](λ). Becauseu2is Parter forλrelative toA, we conclude
thatT1is not a downer branch forλatu2relative toAand, thus,u2must be Parter forλrelative to A[T2]. By (1), the vertexu1must be Parter forλrelative toA[T1], which gives a contradiction.
Corollary 3.LetT be a tree onnvertices, A∈S(T )andλbe an eigenvalue ofA.Letu1and u2be two adjacent Parter vertices forλrelative toA.We denote byT1the component ofT −u2
containingu1and byT2the component ofT −u1containingu2.Thenu1andu2form a Parter
set forλ relative toAif and only ifu1andu2are Parter vertices forλrelative toA[T1]and A[T2],respectively.
Lemma 4.LetT be a tree onnvertices,A∈S(T )andλbe an eigenvalue ofAof multiplicity M(T ).Letu1andu2 be two adjacent Parter vertices forλrelative toA.We denote byT1the
component ofT −u2containingu1and byT2the component ofT −u1containingu2.Then, u1
is a Parter vertex forλrelative toA[T1]andu2is a Parter vertex forλrelative toA[T2],i.e., u1
andu2form a Parter set forλrelative toA.
Proof. Becauseu1andu2are individually Parter vertices forλ, by (2) of Lemma1, we have that mA[Ti](λ)=M(Ti),i=1,2. In order to obtain a contradiction, we suppose thatu1andu2do not
forλrelative toA[Ti], i.e.,mA[Ti−ui](λ)=mA[Ti](λ)−1=M(Ti)−1,i=1,2. We first note
that neitherT1norT2may be a path. If, for example,T2were a path, any component ofT2−u2 would be a path (havingλas an eigenvalue by Lemma1, part (2)) and, hence, these paths should still be downer branches forλatu2relative toA[T −u1], which implies thatu1andu2form a Parter set. Therefore, we may assume thatmA[Ti](λ) >1,i=1,2.
Sinceu1is Parter forλ, we have thatmA(u1)(λ)=M(T )+1. BecauseA(u1)=A[T1−u1] ⊕ A[T2] and under the hypothesis thatu1 and u2 do not form a Parter set implies that u1 is a downer vertex for λ relative toA[T1], it follows that M(T )+1=M(T1)−1+M(T2), i.e., M(T )=M(T1)+M(T2)−2. SincemA[Ti](λ) >1,i=1,2, there must exist a FPS ofkivertices
ofTi whose removal fromTi leavesM(Ti)+ki paths in which, each of the corresponding direct summands of Ahas λas an eigenvalue of multiplicity 1. By hypothesis, ui is not Parter for λrelative toA[Ti],i=1,2, so we may conclude thatu1is a vertex of one pathT′ among the M(T1)+k1paths andu2is a vertex of one pathT′′among theM(T2)+k2paths. If we connectT′ andT′′by the edge{u1, u2}we obtain a subtreeT′′′ofT. Now, consider a matrixB′′′∈S(T′′′) such that λ is an eigenvalue ofB′′′ and change A[T′′′] toB′′′ in Ato obtain B∈S(T ). By construction, we have M(T1)+M(T2)+k1+k2−1 direct summands of B having λ as an eigenvalue, after the removal ofk1+k2vertices and, by the interlacing inequalities, it implies thatmB(λ)M(T1)+M(T2)−1, i.e.,mB(λ) > M(T ), which is a contradiction.
Lemma 5. LetT be a tree onnvertices andA∈S(T ).Suppose thatvis a vertex ofT and that uis a neighbor ofvwhose branch atv, T′,satisfiesmA[T′](λ)=M(T′).Then,eitheruis Parter
forλinT′orvis Parter forλinT (or both).
Proof. Suppose thatuisnotParter forλinT′. IfT′is a path, thenuis a downer inT′, either becauseuis pendent inT′or becauseuis interior inT′and not Parter. Sinceuis a downer forv, vmust be Parter forλ. IfT′is not a path, thenM(T′) >1 and there is a fully FPS forλinT′that does not containu. This is a Parter set forλinT′; call itQ.Qis also a Parter set inT because, by Lemma1, every Parter vertex must have a downer branch not in the direction ofu. The path ofT′in whichuoccurs (and in whichλmust occur) is pendent atvafter removal of the setQ
fromT to produce a treeT′′, which again means thatvis Parter forλinT′′and thatQ∪ {v}is a Parter set forλinT. Thus,vis Parter forλinT.
4. Main results
Theorem 6. Suppose thatT is a tree onnvertices and thatA∈S(T )andλis an eigenvalue of Aof multiplicityM(T ).Then,no vertex ofT is neutral forλ, A.
Proof. Suppose thatλ∈σ (A),A∈S(T )andmA(λ)=M(T ). Then, suppose thatvis a neutral vertex inT forλ,Aand thatT1, . . . , Tkare the branches ofT atvandui is the neighbor ofvin Ti,i=1, . . . , k. Then,
M(T )=mA(λ)=mA(v)(λ)=
k
i=1
mA[Ti](λ)
k
i=1 M[Ti].
By interlacing, either k
i=1
M[Ti] =M(T ) (Case 1) or k
i=1
In Case 1, we reach a contradiction as follows. By Lemma5uiis Parter forλinTi,i=1, . . . , k (asmA[Ti](λ)=M(Ti),i=1, . . . , k) becausevis not Parter forλinT. Thus, since eachuihas
a downer branch forλinTi, each has a downer branch forλinT −vand, thus,{u1, . . . , uk}is a Parter set forλinT −v. Becausevis neutral, we have
mA({u1,...,uk,v})(λ)=M(T )+k=mA({u1,...,uk})(λ).
SinceT − {u1, . . . , uk}containsv, we may define another matrixB∈S(T )such thatB[v] =λ, and, otherwise,Bagrees withA. Now,
mB({u1,...,uk})(λ)=
k
i=1
mA[Ti](λ)+k+1=
k
i=1
M(Ti)+k+1=M(T )+k+1,
which impliesmB(λ)M(T )+1, by interlacing, a contradiction, asB∈S(T ).
In Case 2, we consider two possibilities:k=1, i.e.,vis pendent (Case 2a) andk >1 (Case 2b).
In Case 2a,T −v=T1, and, asvis neutral,mA(v)(λ)=M(T )=M(T1)−1. Since the path cover number cannot increase with the deletion of a pendent vertex, we then have
M(T1)=1+M(T ) > M(T )=P (T )P (T1)=M(T1),
a contradiction showing that Case 2a cannot occur.
In Case 2b (k >1), after an appropriate numbering, we have
mA[Ti](λ)=M(Ti), i=1, . . . , k−1,
and
mA[Tk](λ)=M(Tk)−1.
By Lemma5,ui is Parter forλinTi,i=1, . . . , k−1, or elsevis Parter forλinT (and, thus, not neutral). Sinceui is Parter inTi forλ,i=1, . . . , k−1,{u1, . . . , uk−1}is a Parter set forλ inT, as verified by the same downer branches (which are branches ofT atui,i=1, . . . , k−1). Based upon this, we may calculate
mA({u1,...,uk−1})(λ)= M(T )+k−1
= [M(T1)+ · · · +M(Tk−1)] +M(Tk)−1+k−1.
But, also, definitionally,
mA({u1,...,uk−1})(λ)= mA[T1](λ)+ · · · +mA[Tk−1](λ)+mA[Tk+v](λ)+k−1
= M(T1)+ · · · +M(Tk−1)+M(Tk+v)+k−1,
the second equality following from Lemma1. Comparing the two expressions formA({u1,...,uk−1})(λ),
we conclude that
M(Tk+v)=M(Tk)−1,
a contradiction, asP (Tk)cannot exceedP (Tk+v).
Proof. Suppose to the contrary thatvis a Parter vertex, of degreek, forλwith only one downer branchT1with vertexu1adjacent tov. By (1) of Lemma1,k2. Consider the other branches Tiatvwithui adjacent tov,i=2, . . . , k.
By (2) of Lemma1,λis an eigenvalue of eachA[Ti]with multiplicityM(Ti). Thus, by Theorem
6, eachui is a downer or Parter forλinTi. Under our hypothesis, fori=2, . . . , k,ui is not a downer and so is Parter in Ti. Since ui is Parter inTi,i=2, . . . , k, it is Parter inT; in fact
{u2, . . . , uk}is a Parter set inT. LetT′be the component ofT − {u2, . . . , uk}that includesv. There,vis pendent and still Parter, asT1is a downer branch, andλis an eigenvalue ofA[T′]of multiplicityM(T′). By Lemma1, this is a contradiction.
Theorem 8. Suppose thatT is a tree onnvertices and thatA∈S(T )andλis an eigenvalue
ofAof multiplicityM(T ).Then,upon removal of a Parter vertex forλ, AinT ,no other vertex changes status.
Proof. Call the removed Parter vertexv. By (2) of Lemma 1, each branch atv must display
maximum multiplicity forλ. Thus, by Theorem6, no branch contains a neutral vertex forλ. Thus, it suffices that any vertexuofT other thanvis Parter after removal ofvif and only if it was Parter before. The implication “Parter after” implies “Parter before” is straightforward. If uis Parter after removal ofv, then{u, v}forms a Parter set inT. It follows thatumust have been initially Parter (by counting based upon the interlacing inequalities).
The implication “Parter before” implies “Parter after” is more subtle and, generally, requires that we are working with maximum multiplicity. Suppose thatu /=vis Parter before removal of the Parter vertex v. Sinceuwas Parter before removal ofv, by Theorem7, it has at least two downer branches. Thus, at least one of them does not includevand remains after the removal of v. Thus,u(by virtue of having a downer branch) remains Parter after removal ofv, completing the proof.
Note that in the context of the above proof, it is possible thatuhave a downer branch containing v, so that having additional downer branches is important.
Corollary 9. IfT is a tree onnvertices andA∈S(T )andλis an eigenvalue ofAof multiplicity M(T ),then the set of all (initially) Parter vertices forλ, AinT is a Parter set of vertices forλ, A
inT .
Proof. As Parter vertices are removed fromT, all initially Parter vertices remain Parter and may be removed to further increase the multiplicity ofλ.
Lemma 10. In any tree on n3 vertices, there exists a vertex with at least two pendent
paths.
Proof. LetT be a tree onn3 vertices. For purposes of this proof, we may assume, without loss of generality, thatT has no vertices of degree 2. The claim is obviously true for a path, and any other tree may be “compressed” (via reverse edge-subdivision) to one without degree 2 vertices and some vertices of degree at least 3 in such a way that the occurrence of pendent paths (now pendent vertices) or the degrees of remaining vertices is not changed.
Since the total degree of all vertices ofT is 2(n−1), we have 3h+p2n−2=2p+2h−2
from which we conclude that
hp−2,
which implies that
p > h.
Since the number of pendent vertices exceeds the number of high-degree vertices, there must be a high-degree vertex upon which at least two vertices are pendent. Thus, the original, uncompressed tree had two pendent paths at the same vertex, and the proof of the claim is complete.
See also [10] for a different proof.
Lemma 11. LetTbe a tree on at least three vertices.Then there exists an RPM set whose vertices may be numberedv1, . . . , vqso that removal ofvi+1fromTi =T − {v1, . . . , vi}leaves paths not
present in the forestTi, i=0, . . . , q−1.
Proof. Apply the previous Lemma10 to obtain a vertexv1 with at least two pendent paths.
According to [7], this vertex may be removed on the way to maximizingp−q. Now, another application of Lemma10to any component of the resultingT1that is not a path produces v2 andT2. Continuing in this way until only paths remain, produces thev1, . . . , vqclaimed in the
lemma.
Theorem 12. Suppose thatT is a tree and that0m < M(T )is an integer.Givenλ∈R,there is anA∈S(T )such thatmA(λ)=mand such that there are neutral vertices inT forλ, A.
Proof. In caseT is a path, even on one vertex, the conclusion is immediate by choosingAwith the smallest eigenvalue greater thanλ.
Otherwise, choose an RPM set ofqverticesv1, . . . , vqof the type given in Lemma11. From the “new” paths inTi (previously pendent inTi−1), choose and identify one asPi,i=1, . . . , q. There is a total ofM(T )+qpaths inTq, includingP1, . . . , Pq. Choosem+qof them, including P1, . . . , Pq, to giveP1, . . . , Pq, Pq+1, . . . , Pq+m. This leavesM(T )−m >0 paths. For each pathPi,i=1, . . . , m+q, construct a matrixAi ∈S(Pi)withλas its smallest eigenvalue. By Perron–Frobenius applied to a diagonal similarity ofαiI−Ai, the smallest eigenvalue of any proper principal submatrix ofAiis greater thanλ. For the otherM(T )−mpaths, choose matrices Ai,i=m+q+1, . . . , M(T )+q, so that the smallest eigenvalue is greater thanλ. Note that each vertex ofPi,i=1, . . . , m+q, is a downer forλ,Aiand, by the interlacing inequalities, for i=m+q+1, . . . , M(T )+q, no principal submatrix ofAi hasλas an eigenvalue.
Now letAbe any matrix inS(T )with principal submatricesA1, . . . , AM(T )+qin the appro-priate positions. Then, inT,{v1, . . . , vq}is a Parter set forλ, because inTi−1,Pi is a downer branch atvi, i=1, . . . , q (T0=T). SincemA[Tq](λ)=m+q, by design,mA(λ) must have
beenm.
Finally, we show that any vertexvthat lies in one of the final pathsPi, other thanP1, . . . , Pm+q, is neutral inT forλ,A. Delete one of these vertices fromT to getT′. Since{v1, . . . , vq}is a Parter set inT′forλ,A(v), for the same reason as before, and sincemA[Tq](v)(λ)=m+q, we
Now, the characterization Theorem 13is just the logical combination of Theorems 6 and 12.
Theorem 13. Let mbe a nonnegative integer and T a tree on n vertices. There is a matrix
A∈S(T )and an eigenvalueλ∈σ (A)such thatmA(λ)=mand such that there are neutral
vertices inT forλ, Aif and only ifm < M(T ).
5. Examples
We give here several examples that serve various purposes: (1) to show how our ideas can be used to precisely classify some vertices; (2) to limit the generality of results that we have proven; and, as a by-product, (3) to illustrate our results.
Among all matricesA∈S(T )that have an eigenvalueλsatisfyingmA(λ)=M(T ), some of the vertices must be downers forλ(including, at least, the pendent vertices), some (perhaps the empty set) must be Parter, and the rest (also, perhaps, the empty set) are ambiguous (downers for some suchAand Parter for others).
This is because of Theorem6and Theorem7. In general, it is difficult to classify vertices of the first and second type (though it can be done), but often it may be done easily.
First, consider the tree
which was, early on, difficult for the determination of multiplicity lists. We haveM(T1)=4= P (T1). IfA∈S(T1)andmA(λ)=4, each of the 6 pendent vertices must be downers, and each of vertices 2, 3, 4 lies in every RPM set (there happens to be only the one: {2, 3, 4}), so that each must be Parter. Vertex 1, with no downer branches, cannot be Parter and, so, must be a downer. Thus, every vertex is unambiguously classified. Further, if the multiplicity list is 4,2,2,1,1 (which occurs [8]), no vertex can be neutral foranyeigenvalue. The assignment must beλto each vertex 1, 5, 6, 7, 8, 9, 10 to achieve mA(λ)=4 andµandτ to each path 5−2−6, 7−3−8 and 9−4−10 to achievemA(µ)=mA(τ )=2. The two multiplicities of 1 must correspond to the largest and smallest eigenvalues, via Perron–Frobenius [2], so that, for them, every vertex is a downer. Since every vertex is easily classified as Parter or downer forµandτ, in this case, no vertex is neutral for any eigenvalue. This limits generalization of Theorem6.
M(T2)=3=P (T2), and ifmA(λ)=3 forA∈S(T2), each pendent vertex is a downer. However each of the subsets{2},{1,3},{1,2},{2,3}or{1,2,3}of{1,2,3}may be Parter, so that no vertex is unambiguously Parter.
In contrast toT1it can happen that, even whenM(T )is attained and all other eigenvalues are as multiple as possible, there still may be a neutral vertex for a multiple eigenvalue (of multiplicity less thanM(T )). Let
M(T )=3, and for the assignment shown, the multiplicity list is 3,2,1,1,1, as concentrated as possible. Yet, the top vertex is neutral for the multiplicity 2 eigenvalue (and a downer for the multiplicity 3 eigenvalue).
Now let
M(T4)=3, andmA(λ)=2 with the assignment shown. Vertices 1, 2 and 3 are all Parter forλ, but{1,2,3}is not a Parter set, in contrast to Theorem8.
We also note that it is possible to have arbitrarily many Parter vertices for an eigenvalue attainingM(T )and have each one have only two downer branches. Consider
If there are 3kvertices, thenM(T )=P (T )=k,mA(λ)=kand each of thekinterior vertices is Parter. However, the two pendent vertices at each are the only downer branches for each Parter vertex.
References
[1] M. Fiedler, A characterization of tridiagonal matrices, Linear Algebra Appl., 2 (1969) 191–197. [2] R. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1985.
[3] C.R. Johnson, A. Leal-Duarte, The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree, Linear and Multilinear Algebra 46 (1999) 139–144.
[4] C.R. Johnson, A. Leal-Duarte, C.M. Saiago, D. Sher, Eigenvalues, multiplicities and graphs, Contemp. Math. 419 (2006) 167–183.
[6] C.R. Johnson, A. Leal-Duarte, C.M. Saiago, The Parter–Wiener theorem: refinement and generalization, SIAM J. Matrix Anal. Appl. 25 (2) (2003) 352–361.
[7] C.R. Johnson, C.M. Saiago, Estimation of the maximum multiplicity of an eigenvalue in terms of the vertex degrees of the graph of a matrix, Electron. J. Linear Algebra 9 (2002) 27–31.
[8] C.R. Johnson, C.M. Saiago, Branch duplication for the construction of multiple eigenvalues in an Hermitian matrix whose graph is a tree, Linear and Multilinear Algebra 56 (4) (2008).
[9] C.R. Johnson, B.D. Sutton, Hermitian matrices, eigenvalue multiplicities, and eigenvector components, SIAM J. Matrix Anal. Appl. 26 (2) (2004) 390–399.
[10] P. Nylen, Minimum-rank matrices with prescribed graph, Linear Algebra Appl. 248 (1996) 303–316.
Further reading
[1] C. Chartrand, L. Lesniak, Graphs and Digraphs, Chapman & Hall, London, 1996.
[2] A. Leal-Duarte, Construction of acyclic matrices from spectral data, Linear Algebra Appl. 113 (1989) 173–182. [3] S. Parter, On the eigenvalues and eigenvectors of a class of matrices, J. Soc. Industrial Appl. Math. 8 (1960) 376–388. [4] G. Wiener, Spectral multiplicity and splitting results for a class of qualitative matrices, Linear Algebra Appl. 61 (1984)
