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Branch duplication for the construction of multipleeigenvalues in an Hermitian matrix whose graph is atreeCharles R. Johnson a; Carlos M. Saiago ba Department of Mathematics, College of William and Mary, Williamsburg, VA23187-8795, USAb Departamento de Matem tica, Faculdade de Ci ncias e Tecnologia daUniversidade Nova de Lisboa, 2829-516 Quinta da Torre, Portugal

First Published: July 2008

To cite this Article: Johnson, Charles R. and Saiago, Carlos M. (2008) 'Branchduplication for the construction of multiple eigenvalues in an Hermitian matrix

whose graph is a tree', Linear and Multilinear Algebra, 56:4, 357 — 380

To link to this article: DOI: 10.1080/03081080600597668URL: http://dx.doi.org/10.1080/03081080600597668

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Linear and Multilinear Algebra, Vol. 56, No. 4, July 2008, 357–380

Branch duplication for the constructionof multiple eigenvalues in an Hermitian

matrix whose graph is a tree

CHARLES R. JOHNSONy and CARLOS M. SAIAGO*z

yDepartment of Mathematics, College of William and Mary,P.O. Box 8795, Williamsburg, VA 23187-8795, USA

zDepartamento de Matematica, Faculdade de Ciencias e Tecnologia daUniversidade Nova de Lisboa, 2829-516 Quinta da Torre, Portugal

Communicated by S. Kirkland

(Received 16 April 2004; revised 30 July 2005; in final form 25 January 2006)

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 branchof T. A method for constructing a larger tree T 0, in which the branch is ‘‘duplicated’’, andan Hermitian matrix A0 whose graph is T 0 is described. The eigenvalues of A0 are all ofthose of A, together with those corresponding to the branch, including multiplicities. Thisidea is applied (1) to give a solution to the inverse eigenvalue problem for stars, (2) to provethat the known diameter lower bound, for the minimum number of distinct eigenvaluesamong 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.

Keywords: Hermitian matrices; Eigenvalues; Inverse eigenvalue problem; Multiplicities; Trees;Branch duplication

Mathematics Subject Classifications: 15A18; O5CO5; O5C5O

1. Introduction

Given an n� n Hermitian matrix A ¼ ðaijÞ, the undirected graph of A, GðAÞ, is thegraph on vertices 1, . . . , n with the edge fi, jg, i 6¼ j, if and only if aij 6¼ 0. Given anundirected graph G, we denote by SðGÞ the set of all Hermitian matrices whosegraph is G; no restriction, other than reality, is placed upon the diagonal entries ofA 2 SðGÞ.

A major open question is the inverse eigenvalue problem (IEP) for a given graph G: ifG has n vertices, exactly which sets of n real numbers (including multiplicities) occur as

*Corresponding author. Email: [email protected]

Linear and Multilinear AlgebraISSN 0308-1087 print/ISSN 1563-5139 online � 2008 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/03081080600597668

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008 the spectrum of A for some A 2 SðGÞ. This problem has received considerable study in

case G is a tree, and, though the solution is known for certain trees [1], the problem is,in general, open. We shall also concentrate, here, upon trees (in which case there is nodifference between real symmetric or complex Hermitian matrices, in any of theproblems or constructions discussed).

Our purpose here is to describe a method for constructing certain (typically, but notnecessarily, multiple) eigenvalues in matrices in SðGÞ, G a given tree with somestructure. For reasons that will become clear, we refer to this method as ‘‘branchduplication’’. We then give two applications of branch duplication: (1) the IEP forarrow matrices (i.e., the case in which G is a star); and (2) to prove, for trees of diameterless than 6, that the minimum number of distinct eigenvalues among matrices in SðGÞ,G a tree, is the diameter. The diameter is known to be a lower bound [2] in general, andthere is a diameter 7 tree for which 8 is the minimum number of distinct eigenvalues.

2. Notation and prior results

Let G be an undirected graph on n vertices. If A ¼ ðaijÞ is a matrix in SðGÞ and� � f1, . . . , ng is an index set, we denote the principal submatrix of A resulting fromdeletion (resp. retention) of the rows and columns � by A(�) (resp. A½��).

At times, we consider graphs on indices other than 1, 2, . . . , n. This happens, forexample, when G0 is a subgraph of G induced by a subset of the vertices of G, orwhen we construct a matrix that will be embedded as a principal submatrix of alarger matrix with a given graph. When this happens, we use similar notation, andthis should not lead to confusion. If G0 is a subgraph of G induced by �, we alsowrite AðG0Þ (resp. A½G0�) instead of A(�) (resp. A½��). When � consists of a singleindex i, we abbreviate AðfigÞ (resp. G� fig) by A(i) (resp. G� i). In the same spirit,we abbreviate A½fig� by A½i�.

If A is a matrix in SðGÞ, the subgraph of G induced by deletion of a vertex v, G� v,corresponds, in a natural way, to A(v). In particular, when G is a tree, A(v) is a directsum whose summands correspond to components of G� v (which we call branches ofG at v), the number of summands (or components) being the degree of v in G (whichwe denote by degG(v)).

Throughout, we consider the case in which G is a tree T. If v is an identified vertex ofT of degree k, we identify the neighbors of v in T as u1, u2, . . . , uk, and we denote thebranch of T resulting from deletion of v and containing ui by Ti, i ¼ 1, 2, . . . , k.

For a matrix A, we denote the spectrum of A by �(A), and we denote thecharacteristic polynomial of A by pA(t).

We shall use expansions of the characteristic polynomial of a matrix A whose graph isa tree T. A useful one, known as the neighbors formula (this expansion appears in [3] and,in [4], a detailed account of several expansions of the characteristic polynomial ispresented in a graph-theoretical form), is obtained when attention is focused upon theedges connecting a particular vertex v to its neighbors u1, . . . , uk in T. We have

pAðtÞ ¼ ðt� avvÞYkj¼1

pA½Tj�ðtÞ �Xkj¼1

javuj j2pA½Tj�uj�ðtÞ

Ykl¼1l 6¼j

pA½Tl�ðtÞ: ð1Þ

(Here, we observe the standard convention that the characteristic polynomial of theempty matrix is identically 1.)

358 C. R. Johnson and C. M. Saiago

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008 Denote the (algebraic) multiplicity of � as an eigenvalue of an Hermitian matrix A by

mAð�Þ. If A is a matrix in SðTÞ whose distinct eigenvalues are �1 < � � � < �r, we associatewith A the r-tuple, q ¼ qðAÞ ¼ ðq1, . . . , qrÞ, in which qi ¼ mAð�iÞ, i ¼ 1, . . . , r. Such anr-tuple is the list of ordered multiplicities of A and we denote by LðTÞ the collectionof lists q that occur as A runs over SðTÞ. For example, if A is a 6� 6 Hermitianmatrix with eigenvalues �2,�1,�1,0,0,3 then qðAÞ ¼ ð1, 2, 2, 1Þ. However, as we shallsee, there is no Hermitian matrix whose graph is a star on 6 vertices having �2,�1,�1,0,0,3 as eigenvalues, as the list ð1, 2, 2, 1Þ of ordered multiplicities does not occur.

A list of multiplicities, listed in nonincreasing order (without respect to the numericalvalues of the underlying eigenvalues) is known as a list of unordered multiplicities.All possible lists of unordered multiplicities for trees on 9 vertices are given at theend of this article. Such lists for 8 and 7 or fewer vertices have appeared previouslyin [1] and [2].

3. Branch duplication for trees

Let T be a tree and v be a vertex of T of degree k with branches T1, . . . ,Tk, andcorresponding neighbors u1, . . . , uk. Note that, if A ¼ ðaijÞ is a matrix in SðTÞ, bypermutation similarity, A is similar to a matrix

(Note that all zero entries in the first row and column are omitted.)Consider the tree �T obtained from T by adding s, s� 1, copies of a branch Tj at v.

In such a case, we say that we perform on T an s-branch duplication of Tj at v(an s-branch duplication of Tj, for short). We denote by ukþ1, . . . , ukþs, (resp.�Tkþ1, . . . , �Tkþs) the new neighbors of v (resp. the new branches at v) in T.In the same spirit, given a matrix A ¼ ðaijÞ in SðTÞ, we denote by �A ¼ ð �aijÞ any

particular matrix in Sð �TÞ obtained from A by satisfying the following requirements:

�A½ �Ti� ¼ A½Ti�, i ¼ 1, . . . , k, and �A½ �Ti� ¼ A½Tj�, i ¼ kþ 1, . . . , kþ s; ð2Þ

�avv ¼ avv; ð3Þ

�avui ¼ avui , i 2 f1, . . . , kg n fjg; ð4Þ

�avuj , �avukþ1 , . . . , �avukþs 2 C n f0g :

j �avuj j2 þ j �avukþ1 j

2 þ � � � þ j �avukþs j2 ¼ javuj j

2: ð5Þ

Construction of multiple eigenvalues in an Hermitian matrix 359

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008 In this event, we say that we perform on A an s-summand duplication of A[Tj] at

v (an s-summand duplication of A[Tj], for short) in order to get the matrix A.

THEOREM 1 Let T be a tree and A be a matrix in SðTÞ. Let T be a tree obtained from Tby an s-branch duplication of Tj at v. Let A be a matrix in Sð �TÞ obtained from A by ans-summand duplication of A[(Tj]T) at v. Then,

p �AðtÞ ¼ pAðtÞ pA½Tj�ðtÞ� �s

:

Proof By (1), the characteristic polynomials of A and A may be written as

pAðtÞ ¼ ðt� avvÞpA½T�v�ðtÞ �Xki¼1

javui j2pA½Ti�ui�ðtÞ

Ykl¼1l 6¼i

pA½Tl�ðtÞ, ð6Þ

p �AðtÞ ¼ ðt� �avvÞp �A½ �T�v�ðtÞ �Xkþsi¼1

j �avui j2p �A½ �Ti�ui�

ðtÞYkþsl¼1l 6¼i

p �A½ �Tl�ðtÞ: ð7Þ

Because of (2)–(4), the characteristic polynomial of A, (7), may be rewritten as

p �AðtÞ ¼ ðt� avvÞpA½T�v�ðtÞ pA½Tj�ðtÞ� �s

�Xki¼1i 6¼j


Ykþsl¼1l 6¼i

p �A½ �Tl�ðtÞ

� j �avuj j2pA½Tj�uj�ðtÞ

Ykþsl¼1l 6¼j


�Xkþsi¼kþ1

j �avui j2p �A½ �Ti�ui�

ðtÞYkþsl¼1l 6¼i

p �A½ �Tl�ðtÞ:

Observe that

�A½ �Ti � ui� ¼ A½Tj � uj�, i ¼ kþ 1, . . . , kþ s,

and, from (5), we have

j �avuj j2 þ

Xkþsi¼kþ1

j �avui j2 ¼ javuj j

2:

Thus,


�Xki¼1i 6¼j


Ykþsl¼1l 6¼i


� javuj j2pA½Tj�uj�ðtÞ

Ykþsl¼1l 6¼j

p �A½ �Tl�ðtÞ,


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008 i.e.,


�Xki¼1


Ykþsl¼1l 6¼i

p �A½ �Tl�ðtÞ:

Since, by (2), we have that

�A½ �Tl� ¼ A½Tl�, l ¼ 1, . . . , k

and

�A½ �Tl� ¼ A½Tj�, l ¼ kþ 1, . . . , kþ s,

it follows that

Ykþsl¼1l 6¼i

p �A½ �Tl�ðtÞ ¼

Ykl¼1l 6¼i

pA½Tl�ðtÞ pA½Tj�ðtÞ� �s

, i ¼ 1, . . . , k:

Therefore,

p �AðtÞ ¼ ðt� avvÞpA½T�v�ðtÞ �Xki¼1


Ykl¼1l 6¼i

pA½Tl�ðtÞ

24

35 pA½Tj�ðtÞ� �s

and, by (6), it follows that

p �AðtÞ ¼ pAðtÞ pA½Tj�ðtÞ� �s

:

Example 2 Consider the matrix

A ¼

0ffiffiffi3p

0 0ffiffiffi3p

01

2

ffiffiffi3p

2

01

20 0

0

ffiffiffi3p

20 0

26666666664

37777777775

whose graph is the following tree T

:


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008 The matrix A has eigenvalues �2, 0, 0, 2. Denote by T 0 the branch of T at vertex 1.

We have

A½T 0� ¼

01

2

ffiffiffi3p

21

20 0ffiffiffi

3p

20 0

26666664

37777775

and A½T 0� has eigenvalues �1, 0, 1.Let T be the tree obtained from T by a single branch duplication of T 0 at vertex 1.

Then T is the tree

:

Let A be a matrix whose graph is T, obtained from A by a single summand duplicationof A½T 0� at vertex 1. Then

�A ¼

0 �a12 0 0 �a15 0 0

�a12 01

2

ffiffiffi3p

20 0 0

01

20 0 0 0 0

0

ffiffiffi3p

20 0 0 0 0

�a15 0 0 0 01

2

ffiffiffi3p

2

0 0 0 01

20 0

0 0 0 0

ffiffiffi3p

20 0

266666666666666666666664

377777777777777777777775

,

in which j �a12j2 þ j �a15j

2 ¼ffiffiffi3p 2

and �a12, �a15 2 C n f0g, has eigenvalues �2,� 1, 0, 0, 0, 1, 2.We should note that, when T is a general (undirected) graph having Tj as a branch

at v (there is a bridge connecting vertex v to the component Tj, i.e., vertex v andcomponent Tj are connected by a single edge) the conclusion of Theorem 1 stillholds. This may be checked using another useful expansion of the characteristicpolynomial, known as the bridge formula (this expansion appears in [3] and [4]).Such an expansion is obtained when attention is focused upon the bridge connectingtwo vertices v and uj. Denoting by Tj the component of T resulting from deletion of


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008 v and containing uj, and denoting by Tv the component of T resulting from deletion of uj

and containing v, we have

pAðtÞ ¼ pA½Tv�ðtÞpA½Tj�ðtÞ � javuj j2pA½Tv�v�ðtÞpA½Tj�uj�ðtÞ: ð8Þ

(Again, we observe the standard convention that the characteristic polynomial of theempty matrix is identically 1.)

4. The IEP for Hermitian arrow matrices

In general, an arrow matrix is a matrix whose entries are zero, except those on thediagonal and in the first row and column. Here, we consider the case of Hermitianarrow matrices whose diagonal entries are free (may be also 0). As may be easilychecked, the graph of an arrow matrix is a star (a tree on n vertices having a vertexof degree n� 1).

An IEP for SðTÞ is the following: given an undirected graph T on n vertices and realnumbers �1, . . . , �n, construct an Hermitian matrix A in SðTÞ such that A has �1, . . . , �nas eigenvalues.

As was shown in [5], when T is a tree, given any real numbers �1 < � � � < �n, thereexists always an Hermitian matrix whose graph is T having �1, . . . , �n as eigenvalues.However, depending on the graph T, a matrix in SðTÞ can have multiple eigenvalues.In fact, the graph of an Hermitian matrix can substantially limit the possiblemultiplicities of the eigenvalues (see, e.g. [6]).

For example, it is well known that, if the graph is a path (a tree in which each vertexhas degree at most 2) the IEP has a solution if and only if all the �’s are distinct. Moregenerally [7], if the graph T is a tree, the maximum multiplicity of an eigenvalue amongmatrices in SðTÞ is the path cover number (the minimum number of vertex disjoint paths,occurring as induced subgraphs of T, that cover all the vertices of T), which can be effi-ciently computed. When T is a tree, it is also a known fact that the smallest and largesteigenvalues of any matrix A in SðTÞ each have multiplicity 1 and, moreover, the smallestand largest eigenvalues of A cannot occur as an eigenvalue of a principal submatrix of Aof size one smaller [8].

So, given a graph T, the knowledge of what lists of multiplicities, ordered by thenumerical values of the distinct underlying eigenvalues, which may occur amongmatrices in SðTÞ, is a main key to solve the IEP for SðTÞ.

In [6], the collection of the possible lists of multiplicities that can occur amongmatrices whose graph is a star was completely determined. More recently, in [1], theIEP for SðTÞ when T is a generalized star (a tree having at most one vertex of degreegreater than 2) was solved and an explicit way to construct a matrix in SðTÞ with aprescribed spectrum was given. The answer shows that, at least when T is a generalizedstar, the IEP for SðTÞ is equivalent to determining the collection of the possible lists ofmultiplicities that can occur among matrices in SðTÞ. A star is a particular case of ageneralized star and here we present a somewhat different and easier way to solvethe IEP for SðTÞ, T being a star. This IEP result for the particular case of a star maybe phrased as follows.


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008 LEMMA 3 Let T be a star on n vertices and �1 < �2 < � � � < �r be any sequence of real

numbers. Then there is a matrix in SðTÞ with distinct eigenvalues �1 < �2 < � � � < �rand list of ordered multiplicities ðq1, q2, . . . , qrÞ if and only if ðq1, q2, . . . , qrÞ satisfies thefollowing conditions:

(a)Pr

i¼1 qi ¼ n; and(b) if qi > 1 then 1 < i < r and qi�1 ¼ 1 ¼ qiþ1.

Observe that, if T is a star on n vertices then there is a vertex v whose removal from Tleaves n� 1 isolated vertices (we call such a vertex the central vertex of T). As is wellknown, there is an important relationship between the eigenvalues of a matrix A inSðTÞ and those in A(v), given by the interlacing inequalities for the eigenvalues of anHermitian matrix. Much more can be said for the case of Hermitian matrices whosegraph is a tree (see, e.g. [1,8]). We shall record here a key result for the particularcase in which T is a star.

LEMMA 4 Let T be a star with central vertex v. If A is a matrix in SðTÞ and � is aneigenvalue of A(v) then mAðvÞð�Þ ¼ mAð�Þ þ 1.

We may observe as an immediate consequence of Lemma 4 that, if � is a simpleeigenvalue of A(v) (with multiplicity 1) then � is not an eigenvalue of A. In the sameway, we may conclude that � is an eigenvalue of A of multiplicity m� 2 (a multipleeigenvalue) if and only if � is an eigenvalue of A(v) of multiplicity mþ 1.

A way to construct an Hermitian matrix A in SðTÞ with prescribed spectrum, T beinga star with central vertex v, will be given in this section.

As mentioned earlier, the IEP for a star is a particular case of the IEP for ageneralized star. However, when T is a star, branch (summand) duplication simplifiesthe way to construct an Hermitian matrix in SðTÞ with prescribed spectrum. Whenthe prescribed spectrum has eigenvalues with multiplicities greater than 1, we reducethe problem of constructing a matrix A in SðTÞ with the desired spectrum, to one ofconstructing a matrix A0 in SðT0Þ, in which T0 is a subtree of T containing v, havingas eigenvalues all the simple eigenvalues of the prescribed spectrum for A, each onewith multiplicity 1. Using a convenient choice of the eigenvalues of A0(v) and branch(resp. summand) duplication at v in T0 (resp. in A0), we obtain a matrix A in SðTÞwith the desired spectrum. Throughout this section, T is a star on n vertices with centralvertex v.

The following result appears in [5] and [9].

LEMMA 5 Let T be a star on n vertices, with central vertex v whose neighbors areu1, u2, . . . , un�1. Let �1, �2, . . . , �n, �1,�2, . . . ,�n�1 be real numbers such that

�1 < �1 < �2 < � � � < �n�1 < �n:

Then there is a symmetric matrix A ¼ ðaijÞ in SðTÞ such that

avv ¼Xni¼1

�i �Xn�1i¼1

�i and avuk ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�

Qni¼1ð�k � �iÞQn�1i¼1i 6¼kð�k � �iÞ

vuut , k ¼ 1, . . . , n� 1:


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008 By Lemma 5, an Hermitian matrix A in SðTÞ may be constructed with prescribed

spectrum f�1, �2, . . . , �ng, in which �1 < �2 < � � � < �n. For this purpose, it sufficesto prescribe any real numbers �1 < �2 < � � � < �n�1, such that �1 < �1 <�2 < � � � < �n�1 < �n, as the eigenvalues of A(v).

The construction of an Hermitian matrix A in SðTÞ with prescribed distinct eigen-values �1 < �2 < � � � < �r, and list of ordered multiplicities ðq1, q2, . . . , qrÞ in LðTÞ,in which qi1 , qi2 , . . . , qih (h� 1) are the multiplicities greater than 1, may be carriedout following the procedure described below.

Let s be the number of qi’s in ðq1, q2, . . . , qrÞ such that qi¼ 1. Thus,s ¼ n� ðqi1 þ qi2 þ � � � þ qih Þ and, since ðq1, q2, . . . , qrÞ 2 LðTÞ, by Lemma 3, if qj>1,then qj�1 ¼ 1 ¼ qjþ1. This implies (under our current assumption that h is at least 1)that 1 � h < s.

Let �1 < �2 < � � � < �s be all the �i’s such that qi¼ 1, i ¼ j1, j2, . . . , js. By Lemma 5,considering any sequence of real numbers �1 < �2 < � � � < �s�1 containing�i1 , �i2 , . . . , �ih , and such that �i < �i < �iþ1, i ¼ 1, 2, . . . , s� 1, then there is aHermitian matrix A0 in SðT0Þ, in which T0 is a subtree of T containing v, with�ðA0Þ ¼ f�1,�2, . . . ,�sg and �ðA0ðvÞÞ ¼ f�1,�2, . . . ,�s�1g.

Suppose, without loss of generality, that the k-th diagonal entry of A0(v) is �ik ,k ¼ 1, 2, . . . , h.

Let �A1 be a matrix obtained from A0 by a qi1 -summand duplication of A0½1� at v.By construction, Gð �A1Þ ¼ �T1 in which �T1 is a star (a subtree of T) on sþ qi1 vertices.By Theorem 1, the distinct eigenvalues of �A1 are �1,�2, . . . ,�s, �i1 , with m �A1

ð�i1Þ ¼ qi1 .Repeating this process, we get an Hermitian matrix �Ah, in which �Ah is obtained from�Ah�1 by a qih -summand duplication of �Ah�1½h� at v. By construction, Gð �AhÞ ¼ �Th

in which �Th is a star on sþ qi1 þ � � � þ qih vertices. Since sþ qi1 þ � � � þ qih ¼ n, wehave Gð �AhÞ ¼ T. By Theorem 1, the distinct eigenvalues of �Ah are �1,�2, . . . ,�s, �i1 ,�i2 , . . . , �ih , with m �Ah

ð�ikÞ ¼ qik , k ¼ 1, 2, . . . , h. Because �1,�2, . . . ,�s are the �i’ssuch that qi¼ 1, i ¼ j1, j2, . . . , js, it follows that �1 < �2 < � � � < �r, are the distincteigenvalues of �Ah and ðq1, q2, . . . , qrÞ is the list of ordered multiplicities of �Ah.Note that the requirement that �i1 , �i2 , . . . , �ih are among the �1,�2, . . . ,�s�1 insuresthat ‘‘branches’’ for the multiple eigenvalues are available for duplication.

Example 6 Let T be a star on five vertices with central vertex 1. By Lemma 3, weconclude that ð1, 2, 1, 1Þ 2 LðTÞ. Considering, for example, the real numbers1, 3, 3, 5, 9, we would like to construct an Hermitian matrix A in SðTÞ having prescribedeigenvalues 1, 3, 3, 5, 9 and list of ordered multiplicities ð1, 2, 1, 1Þ. The prescribedeigenvalues of A of multiplicity 1, are �1 ¼ 1, �2 ¼ 5 and �3 ¼ 9. The eigenvalue 3 musthave multiplicity 2 so, by Lemma 4, it must be an eigenvalue of A(1) of multiplicity 3.Setting �1 ¼ 3 and choosing, for example, the real number 7 to be �2, we have�1 < �1 < �2 < �2 < �3 (i.e., 1 < 3 < 5 < 7 < 9). By Lemma 5, we can construct anHermitian matrix A0 whose graph is a subtree of T, and such that A0 has eigenvalues1, 5, 9 and A0ð1Þ has eigenvalues 3, 7. For example, the matrix

A0 ¼

5ffiffiffi6p ffiffiffi

6p

ffiffiffi6p

7 0ffiffiffi6p

0 3

26664

37775


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008 has the desired spectrum. Now, let

�A1 ¼

5ffiffiffi6p

�a13 �a14 �a15ffiffiffi6p

7 0 0 0

�a13 0 3 0 0

�a14 0 0 3 0

�a15 0 0 0 3

26666664

37777775

be a matrix obtained from A0 by a 2-summand duplication of A½3� at vertex 1.Recall that, in this event, from (5), we have

�a13, �a14, �a15 2 C n f0g,

and

j �a13j2 þ j �a14j

2 þ j �a15j2 ¼

ffiffiffi6p 2

:

Choosing, for example, �a13 ¼ 1þ i and �a14 ¼ �a15 ¼ffiffiffi2p

we get

�A1 ¼

5ffiffiffi6p

1þ iffiffiffi2p ffiffiffi

2p

ffiffiffi6p

7 0 0 0

1� i 0 3 0 0ffiffiffi2p

0 0 3 0ffiffiffi2p

0 0 0 3

26666664

37777775:

By construction, Gð �A1Þ ¼ T and, by Theorem 1, �A1 has eigenvalues 1, 3, 3, 5, 9 and,ð1, 2, 1, 1Þ is the list of ordered multiplicities of �A1. Setting A ¼ �A1 we get the desiredmatrix.

5. Trees with diameter less than 6

It was shown in [2] that, for any tree T, the minimum number of distinct eigenvalues ofa matrix in SðTÞ is, at least, the number of vertices in a longest path of T, the diameter ofT, d(T) (usually, the diameter is defined as the number of edges in a longest path of T).The following tree


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008 has diameter 7, but (by process of elimination) its shortest unordered multiplicity

list may easily be shown to be (4,3,2,2,2,1,1,1) with eight distinct eigenvalues(relayed by L. Hogben from discussions with S. Fallat and verified by the authors).

Here, we prove that, at least when the diameter of a tree T is less than 6, there alwaysexists a matrix in SðTÞ with exactly d(T) distinct eigenvalues.

LEMMA 7 Let T be a tree with diameter 5. Then T can be constructed inductively fromthe tree

by duplicating branches at some vertices.

Proof Let T be a tree of diameter 5 and U ¼ fv1, v2, v3, v4, v5g be a subset of the verticesof T such that, the subgraph of T induced by U is the following path T 0

Suppose that there are no pendant vertices adjacent to vertex v3 in T. Let T1 be thebranch of T 0 at v2 on vertex fv1g, T5 be the branch of T 0 at v4 on vertex fv5g and, T2

be the branch of T 0 at v3 on vertices fv1, v2g. Let T00 be the tree obtained from T 0, by

duplication of branches T1, T2 and T5 (for each such branch duplication, the diameterremains 5):

(1) ½degTðv3Þ � degT 0 ðv3Þ�-branch duplication of T2 at v3;(2) ½degTðv2Þ � degT 0 ðv2Þ�-branch duplication of T1 at v2;(3) ½degTðv4Þ � degT 0 ðv4Þ�-branch duplication of T5 at v4.

By construction, T 00 is a subtree of T in which degT 00 ðviÞ ¼ degTðviÞ, i ¼ 1, . . . , 5, and, thevertices adjacent to v2 or v4 (and distinct from v3) in T 00 and T are pendant vertices.Denote by u1, . . . , uk, the vertices adjacent to v3, distinct from v2 and v4, in T and T 00.Let T00i be the branch of T 00 at ui on 1 vertex. Since the diameter of T is 5, to obtain Tfrom T 00 it suffices to perform in T 00 a ½degTðuiÞ � degT 00 ðuiÞ�-branch duplication of T

00

i atui, for i ¼ 1, . . . , k. (We note that, in each case in this proof, the subtracted degree is 2,but we wish to indicate the general strategy.)

If there are pendant vertices adjacent to vertex v3 in T, with a similar process, T canbe obtained from the tree

g

THEOREM 8 Let T be a tree of diameter, d(T), less than 6. Then there is a matrix in SðTÞwith exactly d(T) distinct eigenvalues.

Proof We start with the case in which T is a tree of diameter 5.Let


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008 be a tree T1 and A1 be a matrix in SðT1Þ such that A1½v1� ¼ A1½v3� ¼ A1½v5� ¼ �3 and

�ðA1½fv1, v2g�Þ ¼ �ðA1½fv4, v5g�Þ ¼ f�2, �4g, in which �2 < �3 < �4. Observe that �2, �3and �4 are eigenvalues of A1. In fact, we have mA1ðv2Þð�3Þ ¼ 2, mA1ðv3Þð�2Þ ¼ 2 andmA1ðv3Þð�4Þ ¼ 2. Since the smallest and largest eigenvalues of A1 cannot occur as aneigenvalue of a principal submatrix of A1 of size one smaller, denoting by �1 and �5the smallest and largest eigenvalue of A1, respectively, we have that�1 < �2 < �3 < �4 < �5 are the eigenvalues of A1.

Let

be a tree T2 and A2 be a matrix in SðT2Þ such that A2½v1� ¼ A2½v5� ¼ �3, A2½v6� ¼ �4,�ðA2½fv1, v2g�Þ ¼ �ðA2½fv4, v5g�Þ ¼ f�2, �4g and, �3 2 �ðA2½v3, v6�Þ, in which �2 <�3 < �4. Observe that �2, �3 and �4 are eigenvalues of A2. In fact, we havemA2ðv2Þð�3Þ ¼ 2, mA2ðv3Þð�2Þ ¼ 2 and mA2ðv3Þð�4Þ ¼ 3. Again, neither the smallest nor thelargest eigenvalue of A2 can occur as an eigenvalue of a principal submatrix of A2 ofsize one smaller. Thus, denoting by �1 and �5 the smallest and largest eigenvalueof A2, respectively, we have that �1 < �2 < �3 < �4 < �5 are the distinct eigenvaluesof A2, in which mA2

ð�4Þ ¼ 2.By Lemma 7, any tree T of diameter 5 can be obtained from T1 or T2 by sequential

branch duplication at some vertices, each branch duplication preserving diameter 5.By construction of A1 and A2, if T

0 is a tree of diameter 5 obtained from T1 (or T2)by an s-branch duplication of a given branch T 00, denoting by A0 a matrix obtainedfrom A1 (or A2) by an s-summand duplication of A1½T

00� (or A2½T00�), from

Theorem 1 we conclude that A0 and A1 (or A2) have the same distinct eigenvalues,in number equal 5. Thus, a matrix A in SðTÞ having exactly 5 distinct eigenvalues�1 < �2 < �3 < �4 < �5, can be inductively found from the matrices A1 or A2.

We turn now to the case in which T is a tree of diameter less than 5. If dðTÞ � 2, Tmust be a path and, thus, any matrix in SðTÞ has d(T) distinct eigenvalues. If dðTÞ ¼ 3,then T has at most one vertex v of degree greater than or equal 2. Denoting by n thenumber of vertices of T, if n¼ 3 then T is a path, hence, any matrix in SðTÞ has 3 distincteigenvalues. If n� 4, it is easy to see that the path cover number of T is n� 2 � 2.Since the smallest and largest eigenvalue of a matrix in SðTÞ each have multiplicity 1,it follows that there is a matrix in SðTÞ with exactly three distinct eigenvalues.Now we suppose that T is a tree of diameter 4. Consider the following tree T 0 ofdiameter 4

and a matrix A0 in SðT 0Þ such that A0½v1� ¼ �2, A0½v4� ¼ �3, �3 2 �ðA0½fv1, v2g�Þ and

�2 2 �ðA0½fv3, v4g�Þ. Observe that �2 and �3 are eigenvalues of A0. In fact, we have

mA0ðv2Þð�2Þ ¼ 2 and mA0ðv3Þð�3Þ ¼ 2. Since the smallest and largest eigenvalues of A0

cannot occur as an eigenvalue of a principal submatrix of A0 of size one smaller,denoting by �1 and �4 the smallest and largest eigenvalue of A0, respectively, wehave that �1 < �2 < �3 < �4 are the eigenvalues of A0. Observe that any tree T of


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008 diameter 4 can be obtained from T 0 by sequential branch duplication of the branch on

vertex fv1g at v2 and/or sequential branch duplication of the branch on vertex fv4g at v3.Since A0½v1� ¼ �2 and A0½v4� ¼ �3 and �2, �3 are eigenvalues of A0, by Theorem 1,a matrix A in SðTÞ with exactly four distinct eigenvalues �1 < �2 < �3 < �4 can befound from the matrix A0. g

Example 9 Consider the tree T

of diameter 5. In order to get a matrix A whose graph is T, with five distinct eigenvalues,via the procedure used to prove Theorem 8, we may start from the matrix

A1 ¼

0 2 0 0 0

2 0 1 0 0

0 1 0 2 0

0 0 2 0 2

0 0 0 2 0

26666664

37777775:

The matrix A1 has eigenvalues �3, � 2, 0, 2, 3 and GðA1Þ is the path

on five vertices. Moreover, A1½1� ¼ A1½3� ¼ A1½5� ¼ 0 and �ðA1½f1, 2g�Þ ¼ �ðA1½f4, 5g�Þ ¼f�2, 2g so, by the proof of Theorem 8, a matrix A may be constructed inductively,starting from the matrix A1.

We may obtain a matrix

A2 ¼

0 2 0 0 0 0 0

2 0 1 0 0 0 0

0 1 0ffiffiffi2p

0ffiffiffi2p

0

0 0ffiffiffi2p

0 2 0 0

0 0 0 2 0 0 0

0 0ffiffiffi2p

0 0 0 2

0 0 0 0 0 2 0

2666666666664

3777777777775


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008 with eigenvalues �3, � 2, � 2, 0, 2, 2, 3 and graph

:

Finally, we may find the matrix

A ¼

0 0ffiffiffi2p

0 0 0 0 0 0 0

0 0ffiffiffi2p

0 0 0 0 0 0 0ffiffiffi2p ffiffiffi

2p

0 1 0 0 0 0 0 0

0 0 1 0ffiffiffi2p

0 0ffiffiffi2p

0 0

0 0 0ffiffiffi2p

0ffiffiffi2p ffiffiffi

2p

0 0 0

0 0 0 0ffiffiffi2p

0 0 0 0 0

0 0 0 0ffiffiffi2p

0 0 0 0 0

0 0 0ffiffiffi2p

0 0 0 0ffiffiffi2p ffiffiffi

2p

0 0 0 0 0 0 0ffiffiffi2p

0 0

0 0 0 0 0 0 0ffiffiffi2p

0 0

266666666666666666664

377777777777777777775

with eigenvalues �3, � 2, � 2, 0, 0, 0, 0, 2, 2, 3 and graph T. The list of orderedmultiplicities of A is ð1, 2, 4, 2, 1Þ.

6. Branch duplication. A generalization

Let G be a general (undirected) connected graph and v a vertex of G. Suppose thatfv, u1g, . . . , fv, ukg are k edges whose removal from G leaves two components: G0, theone containing u1, . . . , uk and Gv, the one containing v. We call such a component G0

a k-branch of G at v. (Observe that, when G is a tree and v is a vertex of G, eachbranch of G at v is a 1-branch.)

Without loss of generality, if A ¼ ðaijÞ 2 SðGÞ, we may suppose that A appears as


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008 in which A[Gv] and A½G0� denote the principal submatrices of A whose graphs

are Gv and G0, respectively. Denote by A½G0�ðui, ujÞ the submatrix of A whosegraph is G0 without row ui and column uj of A. By expanding pA(t) along row v of Awe obtain

pAðtÞ ¼ pA½Gv�ðtÞpA½G0 �ðtÞ þ pA½Gv�v�ðtÞ

� avu1Xki¼1

auivð�1Þu1þui�1pA½G0�ðui, u1ÞðtÞ

þ avu2Xki¼1

auivð�1Þu2þui�1pA½G0 �ðui, u2ÞðtÞ þ � � �

þ avukXki¼1

auivð�1Þukþui�1pA½G0 �ðui, ukÞðtÞ

!,

i.e.,

pAðtÞ ¼ pA½Gv�ðtÞpA½G0 �ðtÞ

þ pA½Gv�v�ðtÞXkj¼1

avujXki¼1

auivð�1Þujþui�1pA½G0 �ðui, ujÞðtÞ:

ð9Þ

(Again, we observe the standard convention that the characteristic polynomial ofthe empty matrix is identically 1.)

Note that the neighbors formula for trees (1) and the bridge formula (8) areparticular cases of formula (9).

Our generalization of branch duplication for general graphs is then the following.Consider a graph G obtained from G by adding a copy G00 of a k-branch G0 of G

at vertex v. In such a case, we say that we perform on G a branch duplication of thek-branch G0 at v (a branch duplication of G0, for short). If u1, u2 . . . , uk are the verticesin G0 that are neighbors of v, we denote by ukþ1, ukþ2 . . . , ukþk the new neighbors of vin G (i.e., the vertices in G00 that are neighbors of v in G).

In the same spirit, given a matrix A ¼ ðaijÞ in SðGÞ, we denote by �A ¼ ð �aijÞ anyparticular matrix in Sð �GÞ obtained from A by satisfying the following requirements:

�A½Gv� ¼ A½Gv� and �A½G0� ¼ �A½G00� ¼ A½G0�; ð10Þ

For i ¼ 1, . . . , k, set

�avui ¼ avuiffiffifficp

�avukþi ¼ avuiffiffiffiffiffiffiffiffiffiffiffi1� cp

(, in which 0 < c < 1: ð11Þ

In this event, we say that we perform on A a summand duplication of A½G0� at v(a summand duplication of A½G0�, for short) in order to get the matrix A.


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008 THEOREM 10 Let G be a general (undirected) graph and A be a matrix in SðGÞ. Suppose

that G0 is a k-branch of G at vertex v and let G be a graph obtained from G by a branchduplication of G0 at v. Let A be a matrix in Sð �GÞ obtained from A by a summandduplication of A½G0� at v. Then

p �AðtÞ ¼ pAðtÞpA½G0 �ðtÞ:

Proof By construction, we may suppose that A appears as

Because of (10) and applying formula (9), we may conclude that

p �AðtÞ ¼�pA½Gv�ðtÞpA½G0�ðtÞ þ pA½Gv�v�ðtÞ

�Xkj¼1

�avujXki¼1

�auivð�1Þujþui�1pA½G0 �ðui, ujÞðtÞ

�pA½G0 �ðtÞ

þ pA½Gv�v�ðtÞpA½G0 �ðtÞXkþkj¼kþ1

�avujXkþki¼kþ1

�auivð�1Þujþui�1p �A½G00 �ðui, ujÞ

ðtÞ:

ð12Þ

Observe that, by (10), �A½G00� ¼ A½G0� and, by construction of A, we have that

Xkþkj¼kþ1

�avujXkþki¼kþ1

�auivð�1Þujþui�1p �A½G00�ðui, ujÞ

ðtÞ

¼Xkj¼1

�avukþjXki¼1

�aukþivð�1Þujþui�1pA½G0�ðui, ujÞðtÞ:


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008 Taking into account (11), we may rewrite (12) as

p �AðtÞ ¼�pA½Gv�ðtÞpA½G0 �ðtÞ þ pA½Gv�v�ðtÞ

� cXkj¼1

avujXki¼1

auivð�1Þujþui�1pA½G0�ðui, ujÞðtÞ

�pA½G0 �ðtÞ

þ pA½Gv�v�ðtÞpA½G0�ðtÞð1� cÞXkj¼1

avujXki¼1


¼

�pA½Gv�ðtÞpA½G0 �ðtÞ þ pA½Gv�v�ðtÞ

�Xkj¼1

avujXki¼1


�pA½G0 �ðtÞ,

i.e.,

p �AðtÞ ¼ pAðtÞpA½G0 �ðtÞ: g

We note that the strategy for proving the generalization (Theorem 10) differsfrom that for Theorem 1 by using another determinantal expansion. This presentsboth a generalization and alternate proof of Theorem 1.

Example 11 Consider the matrix

A ¼

1 1 1

1 1 1

1 1 1

264

375

whose graph is G

:

The matrix A has eigenvalues 0, 0, 3. Denote by G0 the 2-branch of G at vertex 1.We have

A½G0� ¼1 1

1 1

� �

and A½G0� has eigenvalues 0, 2. Let G be the graph obtained from G by a branch dupli-cation of G0 at vertex 1. Then G is the graph

:


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008 Let A be a matrix whose graph is G, obtained from A by a summand duplication

of A½G0� at vertex 1. Then

�A ¼

1ffiffifficp ffiffiffi

cp ffiffiffiffiffiffiffiffiffiffiffi

1� cp ffiffiffiffiffiffiffiffiffiffiffi

1� cp

ffiffifficp

1 1 0 0ffiffifficp

1 1 0 0ffiffiffiffiffiffiffiffiffiffiffi1� cp

0 0 1 1ffiffiffiffiffiffiffiffiffiffiffi1� cp

0 0 1 1

26666664

37777775,

in which 0 < c < 1, has eigenvalues 0, 0, 0, 2, 3. Observe that, if c 2 R n ½0, 1�, A isa complex symmetric matrix whose graph is G with eigenvalues 0, 0, 0, 2, 3.

Finally, we note that, in the end, Hermicity/symmetry is not essential for the strategyof branch duplication indicated in Theorem 10.

7. Unordered multiplicity lists for trees on nine vertices

We close by giving all the possible lists of unordered multiplicities for trees on ninevertices (figure 1). For this purpose, it is fundamental the recognition of very specialvertices of a given particular tree T related with multiple eigenvalues of matrices inSðTÞ. By the ‘‘Parter–Wiener Theorem’’ (see, e.g. [8]), when T is a tree and � is an eigen-value of multiplicity m� 2 of a matrix A 2 SðTÞ, there exists a vertex v such thatmAðvÞð�Þ ¼ mAð�Þ þ 1 and � is an eigenvalue of at least three direct summands ofA(v). For historical reasons [3], such a special vertex v is called a Parter vertex (for �,A and T). (See [8], in particular Theorem 8, for a characterization of Parter verticesand related issues.)

In [1], the IEP for generalized stars was solved and LðTÞ was characterized, T being ageneralized star. If a given generalized star T has a vertex of degree greater than 2, we callsuch a vertex the central vertex of T. Our definition of generalized star also includesa path as a (degenerate) special case, in which any vertex may be considered as a centralvertex. Note that, the removal of the central vertex of a generalized star leaves only paths.

Given two generalized stars, T1 and T2, a double generalized star is a tree resultingfrom joining a central vertex of T1 to a central vertex of T2 by an edge. Observe thatthe generalized stars are also (degenerate) double generalized stars. In [1], was alsogiven a characterization of LðTÞ when T is a double generalized star.

We note that 32 of the 47 nine-vertex trees in figure 1 are double generalized stars(or degenerate double generalized stars). The construction of symmetric matrices,whose graph is a generalized star, with prescribed spectrum, together with thecharacterization of LðTÞ when T is a double generalized star allow us to tabulate alllists of unordered multiplicities for trees on nine vertices (using some of the restrictionsdiscussed herein with known necessary conditions as the existence of Parter vertices foreach multiple eigenvalue). Branch duplication was also used to construct matrices witha given list of unordered multiplicities. We report here these results. For each tree,p denotes the path cover number and d denotes the diameter. The presented lists ofunordered multiplicities may be constructed using ideas described herein. For thetree T19 in figure 1, the existence of the list of unordered multiplicities ð2, 2, 2, 1, 1, 1Þ


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Figure 1. Trees on nine vertices.


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Figure 1. Continued.


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was justified by the author Johnson and Brian Sutton presenting the following matrixA 2 SðT19Þ,

A ¼

�2 1 1 1 0 0 0 0 0

1 �1 0 0 0 0 0 0 0

1 0 �1 0 0 0 0 0 0

1 0 0 �1

21 1 0 0 0

0 0 0 1 0 0 0 0 0

0 0 0 1 0 1 1 1 0

0 0 0 0 0 1 1 0 0

0 0 0 0 0 1 01

7

ffiffiffi6

7

r

0 0 0 0 0 0 0

ffiffiffi6

7

r0

266666666666666666666664

377777777777777777777775

,

having three eigenvalues of multiplicity 2 (mAð�1Þ ¼ mAð0Þ ¼ mAð1Þ ¼ 2).The possible lists of unordered multiplicities for trees on fewer than nine vertices may

seen in [2] and [10].



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008 Acknowledgement

This work was done within the activities of Centro de Matematica e Aplicacoes daUniversidade Nova de Lisboa.

References

[1] Johnson, C.R., Leal-Duarte, A. and Saiago, C.M., 2003, Inverse eigenvalue problems and lists ofmultiplicities of eigenvalues for matrices whose graph is a tree: the case of generalized stars anddouble generalized stars. Linear Algebra and Its Applications, 373, 311–330.

[2] Leal-Duarte, A. and Johnson, C.R., 2002, On the minimum number of distinct eigenvalues fora symmetric matrix whose graph is a given tree. Mathematical Inequalities and Applications, 5, 175–180.

[3] Parter, S., 1960, On the eigenvalues and eigenvectors of a class of matrices. Journal of the Society forIndustrial and Applied Mathematics, 8, 376–388.

[4] Maybee, J.S., Olesky, D.D., Van Den Driessche, P. and Wiener, G., 1989, Matrices, digraphs, anddeterminants. SIAM Journal on Matrix Analysis and Applications, 10, 500–519.

[5] Leal-Duarte, A., 1989, Construction of acyclic matrices from spectral data. Linear Algebra and itsApplications, 113, 173–182.

[6] Johnson, C.R., Leal-Duarte, A., Saiago, C.M., Sutton, Brian D. and Witt, Andrew J., 2003,On the relative position of multiple eigenvalues in the spectrum of an Hermitian matrix with a givengraph. Linear Algebra and Its Applications, 363, 147–159.

[7] Johnson, C.R. and Leal-Duarte, A., 1999, The maximum multiplicity of an eigenvalue in a matrix whosegraph is a tree. Linear and Multilinear Algebra, 46, 139–144.

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