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Chapter 34. Multiplicity Lists for the Eigenvalues of Symmetric Matrices with a Given Graph

Chapter 34. Multiplicity Lists for the Eigenvalues of Symmetric Matrices with a Given Graph

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34-2



Handbook of Linear Algebra



If a graph G is not connected, then the multiplicity lists for G may be deduced from those of its components via superposition. Also, graphs with many edges admit particularly rich collections of multiplicity

lists. For example, the complete graph admits all multiplicity lists with the given number of eigenvalues,

except the list in which all eigenvalues are the same. For these reasons, a natural beginning for the study

of multiplicity lists for S(G ) or H(G ) is the case in which G = T , a tree. In addition, trees present several

attractive features for this problem, so much of the research in this area, and this chapter, focuses on trees.



34.1



Multiplicities and Parter Vertices



Definitions:

Let G be a simple graph.

For an n×n real symmetric or complex Hermitian matrix A = [ai j ], the graph of A, denoted by G (A),

is the simple graph on n vertices, labeled 1, 2, . . . , n, with an edge between i and j if and only if ai j = 0.

Let S(G ) (respectively, H(G )), denote the set of all n×n real symmetric (respectively, complex Hermitian) matrices A such that G (A) = G (where G has n vertices). No restriction is placed upon the diagonal

entries of A by G , except that they are real.

If G is the subgraph of G induced by β, A(G ) can be used to denote A(β) and A[G ] to denote A[β].

Given a tree T , the components of T \ { j } are called branches of T at j .

If A ∈ H(G ), λ ∈ σ (A), and j is an index such that α A( j ) (λ) = α A (λ) + 1, then j is called a Parter

index or Parter vertex (for λ, A and G ) (where α A (λ) denotes the multiplicity of λ). Some authors refer

to such a vertex as a Parter–Wiener vertex or a Wiener vertex.

If j is a Parter vertex for an eigenvalue λ of an A ∈ H(G ) such that λ occurs as an eigenvalue of at least

three direct summands of A( j ), j is called a strong Parter vertex.

A downer vertex i in a graph G (for λ ∈ σ (A) and A ∈ H(G )) is a vertex i such that α A(i ) (λ) =

α A (λ) − 1.

A downer branch of a tree T at j is a branch Ti at j , determined by a neighbor i of j such that i is a

downer vertex in Ti (for λ and A[Ti ]).

If a branch of a tree at a vertex j is a path and the neighbor of j in this branch is a pendant vertex of

this path, the branch is a pendant path at j .

Facts:

1. If A ∈ H(G ), then trivially A(i ) ∈ H(G \ {i }).

2. [HJ85] (Interlacing Inequalities) If an n×n Hermitian matrix A has eigenvalues λ1 ≤ λ2 ≤ · · · ≤ λn

and A(i ) has eigenvalues βi,1 ≤ βi,2 ≤ · · · ≤ βi,n−1 , then λ1 ≤ βi,1 ≤ λ2 ≤ βi,2 ≤ · · · ≤ βi,n−1

≤ λn , i = 1, . . . , n.

3. If λ is an eigenvalue of an Hermitian matrix A, then α A (λ) − 1 ≤ α A(i ) (λ) ≤ α A (λ) + 1 for

i = 1, . . . , n.

4. If T is a tree, then any matrix of H(T ) is diagonally unitarily similar to one in S(T ).

5. [JLS03a](Parter–Wiener Theorem: Generalization) Let T be a tree and A ∈ S(T ). Suppose that

there exists an index i and a real number λ such that λ ∈ σ (A) ∩ σ (A(i )). Then

r There is in T a Parter vertex j for λ.

r If α (λ) ≥ 2, then j may be chosen so that δ ( j ) ≥ 3 and so that there are at least three

A

T



components T1 , T2 , and T3 of T \ { j } such that α A[Tk ] (λ) ≥ 1, k = 1, 2, 3.



r If α (λ) = 1, then j may be chosen so that there are two components T and T of T \{ j } such

A

1

2



that α A[Tk ] (λ) = 1, k = 1, 2.



6. [JLS03a] For A ∈ S(T ), T a tree, j is a Parter vertex for λ if and only if there is a downer branch

at j for λ.



Multiplicity Lists for the Eigenvalues of Symmetric Matrices with a Given Graph



34-3



7. [JL] Suppose that G is a simple graph on n vertices that is not a tree. Then

r There is a matrix A ∈ S(G ) with an eigenvalue λ such that there is an index j so that α (λ) =

A



α A( j ) (λ) = 1 and α A(i ) (λ) ≤ 1 for every i = 1, . . . , n.



r There is a matrix B ∈ S(G ) with an eigenvalue λ such that α (λ) ≥ 2 and α

B

B(i ) (λ) = α B (λ) − 1,



for every i = 1, . . . , n.



8. [JLSSW03] Let T be a tree and λ1 < λ2 be eigenvalues of A ∈ S(T ) that share a Parter vertex in T .

Then there is at least one λ ∈ σ (A) such that λ1 < λ < λ2 .

9. Let T be a tree and A ∈ S(T ). If T is a pendant path of a vertex j of T , then T is a downer branch

for each eigenvalue of the direct summand A[T ].

10. Let T be a path and A ∈ S(T ). Then each eigenvalue of A has multiplicity 1.

11. Let A be an n×n irreducible real symmetric tridiagonal matrix. Then

r A has distinct eigenvalues.

r In A(i ), there are at most min{i − 1, n − i } interlacing equalities and this number may occur.

r For each interlacing equality that does occur, the relevant eigenvalues must be an eigenvalue (of



multiplicity 1) of both irreducible principal submatrices of A(i ).

See Example 3 below.

Examples:

1. In general, one might expect that in passing from A to A(i ), multiplicities typically decline. However,

Fact 5 is counter to this intuition in the case for trees. A rather complete statement has evolved

through a series of papers ([Par60], [Wie84], [JLS03a]). In particular, Fact 5 says that when T is

a tree and α A (λ) ≥ 2, there must be a strong Parter vertex because, by interlacing, the hypothesis

λ ∈ σ (A) ∩ σ (A(i )) must be satisfied for any i . However, i itself need not be a Parter vertex. Even

when α A (λ) ≥ 2, it can happen that α A( j ) (λ) = α A (λ) + 1 with δT ( j ) = 1 or δT ( j ) = 2 or λ

appears in only one or two components of T \ { j }, even if δT ( j ) ≥ 3. There may, as well, be several

Parter vertices and even several strong Parter vertices. Much information about Parter vertices may

be found in [JLSSW03] and [JLS03a]. Let λ, µ ∈ R, λ = µ, and consider real symmetric matrices

whose graphs are the following trees, assuming that every diagonal entry corresponds to the label

of the corresponding vertex.

r The vertex v is a Parter vertex for λ in real symmetric matrices for which the graph is each of



the trees in Figure 34.1. We also note that, depending on the tree T , several different vertices

of T could be Parter for an eigenvalue of the same matrix in S(T ). The matrices A[T1 ] and

A[T2 ] each have u and v as Parter vertices for λ.



T1



T2

λ



λ



u

λ



T3



v

λ



λ



u



v



λ



v

λ



λ



λ



λ

λ



δT 1 (v ) = 1

αA [T 1 ] (λ) = 2



δT 2 (v ) = 2

αA [T 2 ] (λ) = 2



δT 3 (v ) = 3

αA [T 3 ] (λ) = 2



αA [T 1 − v ] (λ) = 3



αA [T 2 − v ] (λ) = 3



αA [T 3 − v ] (λ) = 3



FIGURE 34.1 Examples of Parter vertices.



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Handbook of Linear Algebra

r Also, depending on the tree T , the same vertex could be



a Parter vertex for different eigenvalues of a matrix in

S(T ). The vertex v is a Parter vertex for λ and µ in a real

symmetric matrix A for which the graph is the tree in

Figure 34.2. Such a matrix A has λ and µ as eigenvalues

with α A (λ) = 2 = α A (µ). Since it is clear that we have FIGURE 34.2 Vertex v is a Parter

α A(v) (λ) = 3 = α A(v) (µ), it means that v is Parter for λ vertex for λ and µ.

and µ.

2. Though a notion of “Parter vertex” can be defined for nontrees, Fact 7 is the converse to Fact 5 that

.

shows that its remarkable conclusions are generally valid only for trees.

Consider the matrix J 3 whose graph is the cycle C 3 (the possible multiplicities for the eigenvalues

of a matrix whose graph is a cycle was studied in [Fer80]), which is not a tree. The matrix J 3 has

eigenvalues 0, 0, 3. Since the removal of any vertex from C 3 leaves a path, we conclude that there is

no Parter vertex for the multiple eigenvalue 0.

3. If a graph G is a path on n vertices, then G is a tree and if the vertices are labeled consecutively,

any matrix in S(G ) is an irreducible tridiagonal matrix. Conversely, the graph of an irreducible

real symmetric tridiagonal matrix is a path. The very special spectral structure of such matrices has

been of interest for some time for a variety of reasons. Two well-known classical facts are that all

eigenvalues are distinct (i.e., all 1s is the only multiplicity list) and, if a pendant vertex is deleted,

the interlacing inequalities are strict. Both statements follow from Fact 5, but more can be gotten

from Fact 5 as well. If A is n × n real symmetric and 1 ≤ i ≤ n, then as many as n − 1 of the

eigenvalues of A(i ) might coincide with some eigenvalue of A. We refer to such an occurrence as

an “interlacing equality.” If a pendant vertex is removed from a path, no interlacing equalities can

occur, but if an interior vertex is removed, interlacing equalities can occur. The complete picture

in this regard may be also be deduced from Fact 5.



34.2



Maximum Multiplicity and Minimum Rank



Definitions:

Let G be a simple graph.

The maximum multiplicity of G , M(G ), is the maximum multiplicity for a single eigenvalue among

matrices in S(G ).

The minimum rank of G is mr(G ) = min A∈S(G ) rank(A).

The path cover number of G , P (G ), is the minimum number of induced paths of G that do not

intersect, but do cover all vertices of G .

(T ) = max[ p − q ] over all ways in which q vertices may be deleted from G , so as to leave p paths.

Isolated vertices count as (degenerate) paths.

The maximum rank deficiency for G is m(G ) = n − mr (G ), where the order of G is n.

Facts:

Let G be a simple connected graph of order n.

1. If G is a path, P (G ) = 1; otherwise P (G ) > 1.

2. There may be many minimum path covers. See Example 2.

3. Maximizing sets of removed vertices (used in the computation of

number of removed vertices is not unique. See Example 3.



(G )) are not unique; even the



Multiplicity Lists for the Eigenvalues of Symmetric Matrices with a Given Graph



34-5



FIGURE 34.3 Two of the forbidden graphs for mr(G ) = 2.



4.

5.

6.

7.

8.



[Fie69] M(G ) = 1 if and only if G is a path.

M(G ) = n − 1 if and only if G is the complete graph K n .

M(G ) = m(G ).

[JL99] For a tree T , M(T ) = (T ) = P (T ) = m(T ).

[JS02] For any tree T , let HT denote the subgraph of T induced by the vertices of degree at least 3.

For a tree T , (T ) can be computed by the following algorithm. See Example 4.

Algorithm 1: Computation of (T )

Given a tree T .

1. Set Q = ∅ and T = T .

2. While HT = ∅:

Remove from T all vertices v of HT such that δT (v) − δ HT (v) ≥ 2

and add these vertices to Q.

3.



(T ) = p − |Q| where p is the number of components (all of which are paths) in T \Q.



9. [JL99], [BFH04] (G ) ≤ M(G ) and (G ) ≤ P (G ).

10. [BFH04], [BFH05] If n ≥ 2, P (K n ) < M(K n ), where K n is the complete graph on n vertices. If G

is unicyclic (i.e., has a unique cycle), then P (G ) ≥ M(G ) and strict inequality is possible.

11. [BFH04] Minimum rank (and, thus, maximum multiplicity) of a graph with a cut vertex can be

computed from the minimum ranks of induced subgraphs.

12. [BHL04] If H is an induced subgraph of G , then mr(H) ≤ mr(G ). Furthermore, mr(G ) =2

(i.e., M(G ) = n − 2) if and only if G does not contain as an induced subgraph one of the following

four forbidden graphs: the path on 4 vertices P4 , the complete tripartite graph K 3,3,3 , the two graphs

shown in Figure 34.3. Other characterizations are also given.



Examples:

1. Considering the tree T in Figure 34.4, we have P (T ) = 2

1

5

(e.g., 1-3-2 and 5-4-6 constitute a minimal path cover of the

4

3

vertices) and, of course, (T ) = 2, as removal of vertex 4

leaves the 3 paths 1-3-2, 5, and 6 (and neither can be improved

2

6

upon). Note that if submatrices A[{1, 2, 3}], A[{5}], and A[{6}]

of A ∈ S(T ) are constructed so that λ is an eigenvalue of each

(this is always possible and no higher multiplicity in any of them FIGURE 34.4 A tree with path cover

is possible), then α A (λ) ≥ 3 − 1 = 2, which is the maximum number 2.

possible.

2. Consider the tree T on 12 vertices in Figure 34.5. It is not difficult to see that the path cover number

of T , P (T ), is 4. However, it can be achieved by different collections of paths. For example, P (T )

can be achieved from the collection of 4 paths of T , 1-2-3, 4-5-6, 7-8 and 12-9-10-11. Similarly,



34-6



Handbook of Linear Algebra



11



1



3



6



10



2



5



9



4



12



8

7



FIGURE 34.5 A tree with path cover number 4.



the paths of T , 1-2-3, 4-5-8-7, 6 and 12-9-10-11, form a collection of vertex disjoint paths (each

one is an induced subgraph of T ) that cover all the vertices of T .

3. Consider the tree T on 12 vertices in Figure 34.5. As we can see in Table 34.1, (T ) can be achieved

for q = 1, 2, 3. When q = 2, there are 3 different sets of vertices whose removal from T leaves 6

components (paths), i.e., p − q = 4.

4. The algorithm in Fact 8 applied to the tree T in Figure 34.6 gives, in one step, a subset of vertices

of T , Q = {v 2 , v 3 , v 4 , v 5 }, with cardinality 4 and such that T \Q has 13 components, each of which

is a path. Therefore, (T ) = 13 − 4 = 9.

Note that, in any stage of the process to determine (T ), we may not choose just any vertex with

degree greater than or equal to 3.



TABLE 34.1



(T ) for the tree T in Figure 34.5



Removed Vertices from T

5

5, 2

5, 9

5, 10

2, 5, 9

2, 5, 10



q

1

2

2

2

3

3



p − q (=

4

4

4

4

4

4



p

5

6

6

6

7

7



v5

v4



v1



v2



v3



FIGURE 34.6 A tree T with



(T ) = 9.



(T ))



Multiplicity Lists for the Eigenvalues of Symmetric Matrices with a Given Graph



34-7



FIGURE 34.7 A tree of diameter 6 for which the minimum number of distinct eigenvalues is 8 > 6 + 1.



34.3



The Minimum Number of Distinct Eigenvalues



Definitions:

Let T be a tree.

The diameter of T , d(T ), is the maximum number of edges in a path occurring as an induced subgraph

of T .

The minimum number of distinct eigenvalues of T , N (T ), is the minimum, among A ∈ S(T ),

number of distinct eigenvalues of A.

Facts:

1. [JL02a] Let T be a tree. Then N (T ) ≥ d(T ) + 1.

2. [JSa2] If T is a tree such that d(T ) < 5, then there exist matrices in S(T ) attaining as few distinct

eigenvalues as d(T ) + 1.

Examples:

1. Since each entry in a multiplicity list represents a distinct eigenvalue, the “length” of a list represents

the number of different eigenvalues. This number can be as large as n (the number of vertices), of

course, but it cannot be too small. Restrictions upon length limit the possible multiplicity lists. Just

as a path has many distinct eigenvalues, a long (chordless) path occurring as an induced subgraph

of a tree forces a large number of distinct eigenvalues.

2. For many trees T , there exist matrices in S(T ) attaining as few distinct eigenvalues as d(T ) + 1.

However, for the tree T in Figure 34.7, d(T ) = 6 and, in [BF04], the authors have shown that

N (T ) = 8 > d(T )+1. It is not known how to deduce the minimum number of distinct eigenvalues

from the structure of the tree, in general.



34.4



The Number of Eigenvalues Having Multiplicity 1



Definitions:

Given a tree T , let U(T ) be the minimum number of 1s among multiplicity lists occurring for T .

Facts:

1. [JLS03a] For any tree T , the largest and smallest eigenvalues of any A ∈ S(T ) necessarily have

multiplicity 1.

2. [JLS03a] For any tree T on n ≥ 2 vertices, U(T ) ≥ 2 and, for each n, there exist trees T for which

U(T ) = 2.

3. Let T be a tree on n vertices. U(T ) ≥ 2N (T ) − n. In particular, U(T ) ≥ 2(d(T ) + 1) − n.



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Handbook of Linear Algebra



Examples:

1. As with the length of lists, it is relatively easy to have many

1s in a multiplicity list. The more interesting issue is how few of

1s may occur among lists for a given tree T . It certainly depends

. . . ..

upon the tree, as the star (see Figure 34.8) may have just two 1s,

while a path (see Figure 34.9) always has as many as the number

FIGURE 34.8 A star.

of vertices.

2. If T has a diameter that is large relative to its number of vertices (a path is an extremal example),

then it may have to have a minimum number of distinct eigenvalues, which forces U(T ) to be much

greater than 2. However, U(T ) may be greater than 2 for other reasons. For example, for the tree T

in Figure 34.10, d(T ) = 4, n = 8, but U(T ) = 3. It is not known how U(T ) is determined by T ,

and it appears to be quite subtle.



34.5



Existence/Construction of Trees with Given Multiplicities



Definitions:

For a given graph G , the collection of all multiplicity lists is denoted by L(G ). If it is not clear from the

context, we will distinguish the unordered lists as Lu (G ) from the ordered lists Lo (G ).

General Inverse Eigenvalue Problem (GIEP) for S(T ): Given a vertex v of a tree T , what are all the

sequences of real numbers that may occur as eigenvalues of A and A(v), as A runs over S(T )?

Inverse Eigenvalue Problem (IEP) for S(T ): What are all possible spectra that occur among matrices

in S(T ), T being a tree?

A tree T has equivalence of the ordered multiplicity lists and the IEP if a spectrum occurs for some

matrix in S(T ) whenever it is consistent with some list of ordered multiplicities of Lo (T ).

Facts:

1. [Lea89] Let T be a tree on n vertices and v be a vertex of T . Let λ1 , λ2 , . . . , λn and µ1 , µ2 , . . . , µn−1

be real numbers. If

λ1 < µ1 < λ2 < · · · < µn−1 < λn ,

then there exists a matrix A in S(T ) with eigenvalues λ1 , λ2 , . . . , λn , and such that, A(v) has

eigenvalues µ1 , µ2 , . . . , µn−1 .

2. For any tree T on n vertices and any given sequence of n distinct real numbers, there exists a matrix



. . .

FIGURE 34.9 A path.



FIGURE 34.10 A tree T on 8 vertices with d(T ) = 4 and U(T ) = 3.



Multiplicity Lists for the Eigenvalues of Symmetric Matrices with a Given Graph



5



7



4

2



34-9



8

3



6

1



9



10



FIGURE 34.11 A tree for which there is not equivalence between ordered multiplicity lists and the IEP.



in S(T ) having these numbers as eigenvalues.

3. Any path has equivalence of the ordered multiplicity lists and the IEP.

4. [BF04] There exists a tree for which the equivalence of the ordered multiplicity lists and the IEP is

not verified. (See Example 1.)

Examples:

1. For remarkably many small trees and many of the families of trees to be discussed in the next

section, the ordered multiplicity lists are equivalent to the IEP. This is not always the case. Extremal

multiplicity lists for large numbers of vertices can force numerical relations upon the eigenvalues,

as in the tree T shown in Figure 34.11.

Let A be the adjacency matrix of T (i.e., the 0,1-matrix in S(T ) with all diagonal entries 0). The

Parter–Wiener Theorem (Fact 5 Section 34.1) guarantees that αA (0) = 4, and likewise guarantees two eigenvalues of multiplicity 2 (the nonzero eigenvalues of A[{2, 3, 4}] = A[{5, 6, 7}] =

A[{8, 9, 10}], so √

the ordered

list

A is√

(1, 2, 4, 2, 1). In fact, direct computation shows

√ multiplicity



√of √

that σ (A) = (− 5, − 2, − 2, 0, 0, 0, 0, 2, 2, 5).

However, it is not possible to prescribe arbitrary real numbers as the eigenvalues with this ordered

multiplicity list. If A = [ai j ] ∈ S(T ) has eigenvalues λ1 < λ2 < λ3 < λ4 < λ5 with multiplicities

1, 2, 4, 2, 1, respectively, then λ1 +λ5 = λ2 +λ4 . The method used to establish this restriction comes

from [BF04]. By the Parter–Wiener Theorem and an examination of subsets of vertices, a11 = λ3 .

The restriction then follows from comparison of the traces of A and A(1).

It is not known for which trees the determination of all possible ordered multiplicity lists is equivalent to the solution of the IEP. Even when the two are not equivalent for some ordered list, they

may be for all other ordered lists.

2. In the construction of multiplicity lists for a tree, it is often useful (and, perhaps necessary) to know

the solution of the GIEP (or some weak form of it) for some of the subtrees of the tree.

It is often more difficult (than giving necessary restrictions) to construct matrices A ∈ S(T ) with

a given, especially extremal, multiplicity list, even when that list does occur. There are three basic

approaches besides ad hoc methods and computer assisted solution of equations. They are

(a) Manipulation of polynomials, viewing the nonzero entries as variables and targeting a desired

characteristic polynomial (see [Lea89] for an initial reference; this method, based on some nice

formulas for the characteristic polynomial in the case of a tree (see, e.g., [Par60], [MOD89]),

can be quite tedious for larger, more complicated trees).

(b) Careful use of the implicit function theorem (initiated in [JSW]).

(c) Division of the tree into understood parts and using the interlacing inequalities to give lower

bounds that are forced to be attained by known constraints (this is along the lines of the

brief discussion in Example 1 Section 34.2 involving (T ), but for larger trees can lead to

complicated simultaneity conditions).

As an example of method (c) and its subtleties (see also Example 2 Section 34.7), consider again

the tree T in Figure 34.4. Since P (T ) = 2, the maximum multiplicity is 2, and because d(T ) = 3,

there must be at least four distinct eigenvalues, two of which have multiplicity 1. This leaves the



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Handbook of Linear Algebra



question of whether the list (2, 2, 1, 1) (which would have to be the ordered list (1, 2, 2, 1)) can

occur. It can, but this is the nontrivial example with smallest number of vertices. Suppose that the

two multiple eigenvalues are λ and µ. We want A ∈ S(T ) with α A (λ) = 2 and α A (µ) = 2. Each

must have a Parter vertex, which must be either vertex 3 or 4. One must be for λ (and not µ) and

the other for µ (and not λ), as two consecutive eigenvalues cannot share a Parter vertex (Fact 8

section 34.1). So assume that 3 is Parter for λ and 4 for µ. Then, we must have A[{1}] = λ = A[{2}]

and λ ∈ σ (A[{4, 5, 6}]); and A[{5}] = µ = A[{6}] and µ ∈ σ (A[{1, 2, 3}]). A calculation

(or other methods) shows this can be achieved simultaneously.



34.6



Generalized Stars



Definitions:

A tree T in which there is at most one vertex of degree greater than two is a generalized star.

In a generalized star, a vertex v is a central vertex if its neighbors are pendant vertices of their branches,

and each branch is a path. (Note that, under this definition, a path is a (degenerate) generalized star, in

which any vertex is a central vertex. When referring to a path as a generalized star, one vertex has been

fixed as the central vertex.)

For a central vertex v of a generalized star T , each branch of T at v is called an arm of T ; the lengths

of an arm are the number of vertices in the arm.

Supposing that v is a central vertex of a generalized star T , with δT (v) = k. Denote by T1 , . . . , Tk its

arms and by l 1 , . . . , l k the lengths of T1 , . . . , Tk , respectively.

A star on n vertices is a tree in which there is a vertex of degree n − 1.

Let u = (u1 , . . . , ub ), u1 ≥ · · · ≥ ub , and v = (v 1 , . . . , v c ), v 1 ≥ · · · ≥ v c , be two nonincreasing

partitions of integers M and N, respectively. If M < N, denote by ue the partition of N obtained from u

appending 1s to the partition u. Note that if M = N, then ue = u.

Facts:

1. A star is trivially a generalized star.

2. If u and v are two nonincreasing partitions of integers M and N, respectively, M ≤ N, such that

u1 + · · · + us ≤ v 1 + · · · + v s for all s (interpreting us or v s as 0 when s exceeds b or c , respectively),

then trivially v majorizes ue , denoted ue v. See Preliminaries.

3. [JLS03b] Let T be a generalized star on n vertices with central vertex v of degree k, l 1 , . . . , l k be

the lengths of the arms T1 , . . . , Tk , and f (x), g 1 (x), . . . , g k (x) be monic polynomials with all their

roots real in which deg f = n, deg g 1 = l 1 , . . . , deg g k = l k . There exists A ∈ S(T ) such that A

has characteristic polynomial f (x) and A[Ti ] has characteristic polynomial g i (x) if and only if

r Each g (x) has only simple roots.

i

r If λ is a root of g (x) · · · g (x) of multiplicity m ≥ 1, then λ is a root of f (x) of multiplicity

1

k



m − 1.



r The roots of f (x) that are not roots of g (x) · · · g (x) are simple and strictly interlace the set of

1

k



roots of g 1 (x) · · · g k (x) (multiple roots counting only once).



4. [JL02b] Let T be a generalized star on n vertices with central vertex of degree s and arm lengths

l 1 ≥ · · · ≥ l s . Then ( p1 , . . . , pr ) ∈ Lu (T ) if and only if

r



r

i =1



pi = n.



r r ≥ l + l + 1.

1

2

r +1

r p = p

.

h

h + 1 = · · · = pr = 1, in which h =

2

r (p , p , . . . , p





(l 1 − 1, . . . , l l 1 − 1).

1

2

r −l 1 −1 )



34-11



Multiplicity Lists for the Eigenvalues of Symmetric Matrices with a Given Graph



T1



T2



v1



T3



v2



v3



FIGURE 34.12 T1 , T2 , and T3 are generalized stars on 9 vertices with central vertices v 1 , v 2 , and v 3 , respectively.



5. [JLS03b] Let T be a generalized star on n vertices with central vertex of degree s and arm lengths

l 1 ≥ · · · ≥ l s . Let λ1 < · · · < λr be any sequence of real numbers. Then there exists a matrix A ∈

S(T ) with distinct eigenvalues λ1 < · · · < λr and list of ordered multiplicities q = (q 1 , . . . , qr ) if

and only if q satisfies the following conditions:

r



r

i =1



q i = n.



r If q > 1, then 1 < i < r and q

i

i − 1 = 1 = qi + 1 .

r (q + 1, . . . , q + 1)

(l 1 , . . . , l s )∗ , in which q i 1 ≥ · · · ≥ q i h are the entries of the r -tuple

i1

ih

e



(q 1 , . . . , qr ) greater than 1.



(That is, when T is a generalized star, there is equivalence of the ordered multiplicity lists and the IEP.)



Examples:

1. Let T1 , T2 , and T3 be the generalized stars in Figure 34.12. We have

Lu (T1 ) = {(1, 1, 1, 1, 1, 1, 1, 1, 1), (2, 1, 1, 1, 1, 1, 1, 1)},

Lu (T2 ) = {(1, 1, 1, 1, 1, 1, 1, 1, 1), (2, 1, 1, 1, 1, 1, 1, 1), (2, 2, 1, 1, 1, 1, 1),

(3, 1, 1, 1, 1, 1, 1), (3, 2, 1, 1, 1, 1), (3, 3, 1, 1, 1), (4, 1, 1, 1, 1, 1),

(4, 2, 1, 1, 1)},

and

Lu (T3 ) = {(1, 1, 1, 1, 1, 1, 1, 1, 1), (2, 1, 1, 1, 1, 1, 1, 1), (2, 2, 1, 1, 1, 1, 1),

(3, 1, 1, 1, 1, 1, 1), (3, 2, 1, 1, 1, 1), (3, 3, 1, 1, 1), (4, 1, 1, 1, 1, 1),

(4, 2, 1, 1, 1), (5, 1, 1, 1, 1), (6, 1, 1, 1), (7, 1, 1)}.



34.7



Double Generalized Stars



Definitions:

A double generalized star is a tree resulting from joining the central vertices of two generalized stars

T1 and T2 by an edge. Such a tree will be denoted by D(T1 , T2 ).

A double star is a double generalized star D(T1 , T2 ) in which T1 and T2 are stars.



34-12



Handbook of Linear Algebra



i1



j1



i2



j2



ik



. . .



. . .



. . .



. . .

jl



i p−1



ip



j q−1



jq



FIGURE 34.13 A double path.



A double path is a double generalized star D(T1 , T2 ) in which T1 and T2 are paths. When we refer to

a double path T on n = p + q vertices we suppose T is represented as in Figure 34.13, in which the

only constraint on the connecting edge {i k , jl } is that not both k ∈ {1, p} and l ∈ {1, q }. The upper (i )

path has k − 1 vertices to the left of the connecting vertex and another p − k vertices to the right; set

s 1 = min{k − 1, p − k}, s 2 = min{l − 1, q − l }, and s = min{q , p, s 1 + s 2 }.

Let G be a tree. Let v be a vertex of G of degree k and G 1 , . . . , G k be the components of G \ {v} having

order l 1 , . . . , l k , respectively. To the tree G is associated the generalized star, Sv (G ), with central vertex

v of degree k, and with arms T1 , . . . , Tk of lengths l 1 , . . . , l k , respectively.

Let u1 and u2 be adjacent vertices of a tree G . Denote by G u1 the connected component of G \{u2 } that

contains u1 and by G u2 the connected component of G \{u1 } that contains u2 . Put S1 = Su1 (G u1 ) and

S2 = Su2 (G u2 ). Now, to the tree G is associated the double generalized star D(S1 , S2 ), which is denoted

by Du1 ,u2 (G ).

Given a vertex v of a tree T and an eigenvalue λ of a matrix A ∈ S(T ), λ is an upward eigenvalue of A

at v if α A(v) (λ) = α A (λ) + 1, and α A (λ) is an upward multiplicity of A at v.

If q = q (A) = (q 1 , . . . , qr ) is the list of ordered multiplicities of A, define the list of upward multiplicities of A at v, denoted by qˆ , as the list with the same entries as q but in which any upward multiplicity

q i of A at v is marked as qˆ i in qˆ .

Given a generalized star T with central vertex v, we denote by Lˆ o (T ) the set of all lists of upward

multiplicities at v occurring among matrices in S(T ).

Facts:

1. A double path is a tree whose path cover number is 2.

2. [JLS03b] Let T be a generalized star on n vertices with central vertex v of degree k and arm lengths

l 1 ≥ · · · ≥ l k . Let λ1 < · · · < λr be any sequence of real numbers. Then there exists a matrix A in

S(T ) with distinct eigenvalues λ1 < · · · < λr and a list of upward multiplicities qˆ = (q 1 , . . . , qr )

if and only if qˆ satisfies the following conditions:

r



r

i =1



q i = n.



r If q is an upward multiplicity in qˆ , then 1 < i < r and neither q

i

i −1 nor q i −1 is an upward



multiplicity in qˆ .

r (q + 1, . . . , q + 1)

i1

ih

e



(l 1 , . . . , l k )∗ , in which q i 1 ≥ · · · ≥ q i h are the upward multiplicities



of qˆ .

3. [JLS03b] (Superposition Principle) Let D(T1 , T2 ) be a double generalized star, bˆ = (b1 , . . . , bs 1 ) ∈

Lˆ o (T1 ), and cˆ = (c 1 , . . . , c s 2 ) ∈ Lˆ o (T2 ). Construct any b + = (b1+ , . . . , bs+1 +t1 ) and c + = (c 1+ , . . . , c s+2 +t2 )

subject to the following conditions:

r t , t ∈ N and s + t = s + t .

1 2

0

1

1

2

2

r b + (respectively, c + ) is obtained from bˆ (respectively, cˆ ) by inserting t (respectively, t ) 0s.

1

2

r b + and c + cannot both be 0.

i

i

r If b + > 0 and c + > 0, then at least one of the b + or c + must be an upward multiplicity of bˆ or cˆ .

i

i

i

i



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