2019/2/23, 中研院, The largest real eigenvalues of nonnegative matrices by applying Kelmans transformations, (2019 Taipei International Workshop on Combinatorics and Graph Theory, 2019/2/20-23)

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The largest real eigenvalues of nonnegative matrices

by applying Kelmans transformations

Chih-wen Weng (joint work with Louis Kao)

Department of Applied Mathematics National Chiao Tung University

14:25-15:10, February 23, 2019

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Undirected graphs

G:

r r r r

r r

u v

a b c d V(G) ={u, v, a, b, c, d}

E(G) ={ua, ub, uc, uv, vc, vd}

N(u) ={a, b, c, v}

N(v) ={u, c, d}

N[v] ={u, c, d, v}

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Adjacency matrices

G : _r _r _r

1 2 3 −→ A(G) =



 0 1 0 1 0 1 0 1 0





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Kelmans transformations for undirected graphs

The graph G(a, b)is obtained from G with the same vertex set and replacing edge ac by bc for each c∈ N(a) − N[b].

r r r r

r r

a b

G : −→ G(a, b) :

r r r r

r r

a b

b a







0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 1

1 1 1 0 1 0







−→







0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0

1 1 1 1 0 1

0 0 1 0 1 0







(unchanged)

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A.K. Kelmans, On graphs with randomly deleted edges, Acta Math. Acad.

Sci. Hung. 37 (1981) 77-88.

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Remark 1: G(a, b) is isomorphic to G(b, a)

r r r r

r r

a b

G : −→ G(a, b) :

r r r r

r r

a b

r r r r

r r

a b

G : −→ G(b, a) :

r r r r

r r

a b

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Complement graph

r r r r

r r

a b

G : −→ G :

r r r r

r r

a b

b a







0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 1 1 1 1 0 1 0







−→







0 1 1 1 1 0 1 0 1 1 1 0 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0







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Remark 2: G(b, a) = G(a, b)

b a







0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 1 1 1 1 0 1 0







−→ G(a, b) =







0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 1 1 1 1 0 1

0 0 1 0 1 0







b a







0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 1 1 1 1 0 1 0







−→ G(b, a) =







0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 1 1 1 0 0 1 0 0 0 0 0 0

1 1 0 1 0 0







=G(a, b)

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Eigenvalues and eigenvectors

Let A be a real square matrix, u is a column vector, λ is a complex number. The number λ is an eigenvalueof A with associatedeigenvector u if

Au = λu.

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A few facts from linear algebra

1 An n× n real symmetric matrix has n real eigenvalues and n corresponding orthonormal eigenvectors.

2 A nonnegative matrix M has a largest real eigenvalue ρ(M), called the spectral radius of M.

3 The eigenvalue ρ(M) is associated with a nonnegative eigenvector of length 1, called Perron vector.

4 All other eigenvalues λ of M satisfy |λ| ≤ ρ(M).

5 Thespectral radius ρ(G) of a graph G is the largest real eigenvalue of its adjacency matrix A = A(G).

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Example

G : _r _r _r

a b c A =



 0 1 0 1 0 1 0 1 0







0 1 0 1 0 1 0 1 0









√2

√±2 2



 = ±√ 2





√2

±2√ 2



 ,



0 1 0 1 0 1 0 1 0







 1 0

−1



 = 0.

Note that ¹

2√ 2(√

2, 2,√

2) is the Perron vector of A and the three eigenvectors

1 2√

2(√ 2, 2,√

2), 1 2√

2(√

2,−2,√ 2), 1

√2(1, 0,−1)

are orthonormal and ρ(G) =√ 2.

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Rayleigh quotient for symmetric matrices

Lemma

Let A = (a_ij) be an n× n symmetric matrix. Then

ρ(A) = max

x̸=0

x^tAx x^tx . Proof.

Let λ₁ ≥ λ2 ≥ · · · ≥ λn be eigenvalues of A with corresponding orthonormal eigenvectors x₁, x₂, . . . , x_n. Write x =∑_n

i=1c_ix_i, where

∑_n

i=1c²_i = 1. Then the maximum of x^tAx =

∑n i=1

c²_iλi

is λ₁ and is when x =±x1.

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Non-symmetric matrices

x:xmax^tx=1x^t (0 1

0 0 )

x = max

x=(a,b) a2+b2=1

ab = 1

2 ̸= 0 = ρ (0 1

0 0 )

.

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Lemma

Let x = (x_a) be the Perron vector of A(G). Then x_b≥ xa ⇒ ρ(G) ≤ ρ(G(a, b)).

Proof.

ρ(G(a, b)) = max

∥y∥²=1

y^tA(G(a, b))y

≥ x^tA(G(a, b))x = ∑

cd∈E(G(a,b))

x_cx_d

= ∑

cd∈E(G)

x_cx_d+ ∑

e∈N(a)−N[b]

x_bx_e− xax_e

≥ x^tAx = ρ(G).

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An old result for undirected graphs

Theorem

ρ(G)≤ ρ(G(a, b)) Proof.

Let x = (x_a) be the Perron vector of A(G). Then

x_b≥ xa ⇒ ρ(G)≤ ρ(G(a, b)), x_a≥ xb ⇒ ρ(G)≤ ρ(G(b, a)), G(a, b)) ∼= G(b, a) ⇒ ρ(G(a, b)) = ρ(G(b, a)).

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Key fact in the above proof

G(a, b) ∼= G(b, a)

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Corollary

ρ(G)≤ ρ(G(a, b)) Proof.

ρ(G)≤ ρ(G(b, a)) = ρ(G(a, b))

The above results ρ(G)≤ ρ(G(a, b)) and ρ(G) ≤ ρ(G(a, b)) are from [Péer Csikvári, On a conjecture of V. Nikiforov, Discrete Mathematics, 309(13), 2009, 4522-4526]

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A counterexample for digraph transformation

A = a b







0 0 1 1 1 0 0 1 1 1 0 0 1 1 1 0





 → A^′ =







0 0 1 1 1 0 1 0 1 1 0 0 1 1 1 0







unchanged

ρ(A)≈ 2.234 > 2.148 ≈ ρ(A^′)

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We will define Kelmans transformations for nonnegative matrices.

Let C denote a nonnegative square matrix of order n. For a subset S of [n] :={1, 2, . . . , n} let C[S] denote the principal submatrix of C restricted to S.

Fix 1≤ a, b ≤ n.

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Kelmans transformation of C from b to a (w.r.t. t

_i

, s

_j

)

C =







c d e c₁₄ c₁₅ f g h c₂₄ c₂₅ i j k c₃₄ c₃₅ c₄₁ c₄₂ c₄₃ ℓ m c₅₁ c₅₂ c₅₃ n o





 a b

≥ 0

→ C^′ =







c d e c₁₄+ t₁ c₁₅− t1

f g h c₂₄+ t2 c₂₅− t2

i j k c₃₄+ t₃ c₃₅− t3

c₄₁+ s₁ c₄₂+ s₂ c₄₃+ s₃ ℓ m c₅₁− s1 c₅₂− s2 c₅₃− s3 n o





≥ 0,

where c_ia+ ti≥ cib− ti and c_aj+ sj ≥ cbj− sj.

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Example

For a = 4, b = 5,t₁= 1, t2 = t3 = 0, s1= s3= 0,s₂ = 1,

C =







c d e 0 1 f g h 1 0 i j k 1 1 1 0 1 ℓ m 0 1 1 n o





 → C^′ =







c d e 1 0 f g h 1 0 i j k 1 1 1 1 1 ℓ m 0 0 1 n o





.

For a = 5, b = 4,t₂= 1, t1 = t3 = 0, s2= s3= 0,s₁ = 1,

C =







c d e 0 1 f g h 1 0 i j k 1 1 1 0 1 ℓ m 0 1 1 n o





 → C^′ =







c d e 0 1 f g h 0 1 i j k 1 1 0 0 1 ℓ m 1 1 1 n o





(= C^′′ later).

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C

^′

and C

^′′

Throughout letC^′ denote the Kelmans transformation of C from b to a with respect to t_i and s_j, and letC^′′= (c^′′_ij) is the Kelmans transformation of C from a to bwith respect to c_ia− cib+ t_i and c_aj− cbj+ s_j.

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Symmetric subgraph

A matrix C is symmetricon{a, b} of if the principal submatrix C[a, b] is symmetric with constant diagonals.

Example The matrix

C =







1 1 0 0 1 0 1 0 1 0 0 1 1 1 1 1 0 1 s t 0 1 1 t s







is symmetric on {4, 5}.

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Remark: C

^′

is similar to C

^′′

If C is symmetric on {a, b}, then C^′ is similar to C^′′.

C^′ =







c d e c₁₄+ t1 c₁₅− t1

f g h c₂₄+ t₂ c₂₅− t2

i j k c₃₄+ t₃ c₃₅− t3

c₄₁+ s₁ c₄₂+ s₂ c₄₃+ s₃ s t c₅₁− s1 c₅₂− s2 c₅₃− s3 t s







C^′′=







c d e c₁₅− t1 c₁₄+ t1

f g h c₂₅− t2 c₂₄+ t₂ i j k c₃₅− t3 c₃₄+ t3

c₅₁− s1 c₅₂− s2 c₅₃− s3 s t c₄₁+ s1 c₄₂+ s2 c₄₃+ s3 t s







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Main result

Theorem

If a nonnegative matrix C is symmetric on {a, b}, then ρ(C) ≤ ρ(C^′).

For the remaining of the talk, we will prove the above theorem.

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Strongly connected

A digraph is strongly connectedif for any two vertices x, y there exists a directed path from x to y.

r r

r

1 2

3 - R

V(G) ={1, 2, 3}

E(G) ={13, 12, 23, 32}

A =



0 1 1 0 0 1 0 1 0





The above digraph is notstrongly connected because there is no directed path from 2 to 1.

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Digraph associated with a matrix

For an n× n matrix M, we define a digraph G(M) with vertex set {1, 2, . . . , n} and edge set

E ={xy | Mxy̸= 0}.

G(M) is called the digraph associated with M.

M =



0 0 −2

3 0 2

0 −1 0



 −→

r r

r

1 2

3

R

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Irreducible matrix

A matrix is irreducibleif its associated digraph is strongly connected.

The matrix

A =



0 1 1 0 0 1 0 1 0





is reduciblesince its associated digraph G(A) is not strongly connected.

r r

r

1 2

3 - R

V(G(A)) ={1, 2, 3}

E(G(A)) ={13, 12, 23, 32}

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A fact from linear algebra

A nonnegative irreducible matrix has a positivePerron vector.

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The irreducible matrix A

^ϵ

For a binary matrix A = (a_ij) and 0 < ϵ < 1, define the matrix A^ϵ = A + ϵJ, where J is the matrix with all 1’s.

A =



0 1 1 0 0 1 0 1 0



 ⇒ A^ϵ =



ϵ 1 + ϵ 1 + ϵ ϵ ϵ 1 + ϵ ϵ 1 + ϵ ϵ





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The idea

Our proof is motivated by the following exercise.

Exercise

Suppose 0≤C≤ C^′. Show ρ(C)≤ ρ(C^′).

Proof.

Let u^t≥ 0 be left Perron vector of C for ρ(C) and v^ϵ> 0 be right Perron vector of C^′ϵ for ρ(C^′ϵ). Then

ρ(C)u^tv^ϵ= u^tCv^ϵ≤u^tC^′ϵv^ϵ= ρ(C^′ϵ)u^tv^ϵ, which implies ρ(C)≤ ρ(C^′ϵ) since u^tv^ϵ > 0. Then by continuity

ρ(C)≤ lim

ϵ→0⁺ρ(C^′ϵ) = ρ(C^′).

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The left Perron vector w

^t

for ρ(C)

Let w^t= (wi) denote the left Perron vector for ρ(C) of C.

We will show that

w_a≥ wb ⇒ ρ(C) ≤ ρ(C^′),

w_b≥ wa ⇒ ρ(C) ≤ ρ(C^′′) = ρ(C^′).

By symmetric, we might assume w_a≥ wb and only prove ρ(C)≤ ρ(C^′).

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The matrix Q = I

_n

− E

^ba

Set v^t= w^tQ, where Q = I_n− E^ba and E^ba denote the binary matrix of order n with a unique 1 in the position ba.

v^t=(w1, w2, w3, w4− w5, w5)

=(w1, w2, w3, w4, w5)







1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 −1 1





 a b

=w^tQ

By the assumption w_a≥ wb, v^t≥ 0

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v

^t

Q

⁻¹

C = w

^t

C = ρ(C)w

^t

= ρ(C)v

^t

Q

⁻¹

v^t= w^tQ and w^tC = ρ(C)w^t

⇒ v^tQ⁻¹C = ρ(C)v^tQ⁻¹

Q⁻¹ =







1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1





 a b

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The matrix Q

⁻¹

C(Q

⁻¹

)

^t

Q⁻¹C(Q⁻¹)^t

=







c₁₁ · · · c_{1 n}₋₁ c_1n₋₁+ c1n

... ... ...

c_n₋₁₁ · · · c_n_−1n−1 c_n_−1n−1+ cn−1n

c_n_{−1 1}+ cn1 · · · cn−1 n−1+ cnn−1 c_n_−1n−1+ cnn−1

+cn−1 n+ cnn





 a b

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Q

⁻¹

C(Q

⁻¹

)

^t

≤ Q

⁻¹

C

^′ϵ

(Q

⁻¹

)

^t

C =







0 0 1 1 1 0 0 1 1 1 0 1 1 1 1 0





 a b

→ C^′ =







0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 0







⇒ Q⁻¹C(Q⁻¹)^t=







0 0 1 2 1 0 0 1 1 1 0 1 2 2 1 2





 ≤







0 0 1 2 1 0 1 1 1 1 0 1 2 2 1 2





 = Q⁻¹C^′(Q⁻¹)^t

<Q⁻¹C^′ϵ(Q⁻¹)^t

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The right eigenvector vector u

^ϵ

for ρ(Q

^t

C

^′ϵ

(Q

^t

)

⁻¹

)

Let u^ϵ̸< 0 denote the right eigenvector vector for ρ(Q^tC^′ϵ(Q^t)⁻¹).

(C^′ϵ is positive, but Q^tC^′ϵ(Q^t)⁻¹ could have some negative entries) Then

(i) ρ(C^′ϵ) = ρ(Q^tC^′ϵ(Q^t)⁻¹);

(ii) C^′ϵ(Q⁻¹)^tu^ϵ = C^′ϵ(Q^t)⁻¹u^ϵ = ρ(C^′ϵ)(Q^t)⁻¹u^ϵ= ρ(C^′ϵ)(Q⁻¹)^tu^ϵ; (iii) (Q⁻¹)^tu^ϵ> 0is an eigenvector for ρ(C^′ϵ).

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Right multiplication of u

^ϵ

Q⁻¹C(Q⁻¹)^tu^ϵ≤Q⁻¹C^′ϵ(Q⁻¹)^tu^ϵ= ρ(C^′ϵ)Q⁻¹(Q⁻¹)^tu^ϵ.

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Left multiplication of v

^t

ρ(C)v^tQ⁻¹(Q⁻¹)^tu^ϵ= v^tQ⁻¹C(Q⁻¹)^tu^ϵ ≤ v^tρ(C^′ϵ)Q⁻¹(Q⁻¹)^tu^ϵ.

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Deleting the positive term v

^t

Q

⁻¹

(Q

⁻¹

)

^t

u

^ϵ

ρ(C)v^tQ⁻¹(Q⁻¹)^tu^ϵ≤ ρ(C^′ϵ)v^tQ⁻¹(Q⁻¹)^tu^ϵ

⇒ ρ(C) ≤ ρ(C^′).

The proof is completed.

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2019/2/23, 中研院, The largest real eigenvalues of nonnegative matrices by applying Kelmans transformations, (2019 Taipei International Workshop on Combinatorics and Graph Theory, 2019/2/20-23)

The largest real eigenvalues of nonnegative matrices

by applying Kelmans transformations

Undirected graphs

Adjacency matrices

Kelmans transformations for undirected graphs

Remark 1: G(a, b) is isomorphic to G(b, a)

Complement graph

Remark 2: G(b, a) = G(a, b)

Eigenvalues and eigenvectors

A few facts from linear algebra

Example

Rayleigh quotient for symmetric matrices

Non-symmetric matrices

An old result for undirected graphs

Key fact in the above proof

Corollary

A counterexample for digraph transformation

Kelmans transformation of C from b to a (w.r.t. t

, s

)

Example

C

and C

Symmetric subgraph

Remark: C

is similar to C

Main result

Strongly connected

Digraph associated with a matrix

Irreducible matrix

A fact from linear algebra

The irreducible matrix A

The idea

The left Perron vector w

for ρ(C)

The matrix Q = I

− E

v

Q

C = w

C = ρ(C)w

= ρ(C)v

Q

The matrix Q

C(Q

)

Q

C(Q

)

≤ Q

C

(Q

)

The right eigenvector vector u

for ρ(Q

C

(Q

)

)

Right multiplication of u

Left multiplication of v

Deleting the positive term v

Q

(Q

)

u

Thank you for your attention.