The Laplacian spectral radius of a graph

The Laplacian spectral radius of a graph

Fan-Hsuan Lin Advisor: Chih-Wen Weng

National Chiao Tung University

August 2, 2014

Outline

1... Introdution

2... Preliminaries

3... Main Results

Some Corollary about Theorem 1 Main Theorem

Applications of the main theorem

4... Conjecture

Outline

1... Introdution

2... Preliminaries

3... Main Results

Some Corollary about Theorem 1 Main Theorem

Applications of the main theorem

4... Conjecture

.Definition ..

Let G = (V, E) be a simple connected graph with vertex set V ={v1, v2,· · · , vn} and edge set E. Let A(G) be the adjacency matrix of G. Denote by d_i=|G1(v_i)| the degree of vertex vi ∈ V (G), where G₁(v_i)is the set of neighbors of v_i, and let

D(G)=diag(d₁, d2,· · · , dn) be the diagonal matrix with entries d₁, d₂,· · · , dn. Then the matrix

L(G) = D(G)− A(G)

is called the Laplacian matrix of a graph G. The Laplacian spectrum of G is

S(G) = (ℓ₁(G), ℓ₂(G),· · · , ℓn(G)),

where ℓ₁(G)≥ ℓ2(G)≥ · · · ≥ ℓn(G)are eigenvalues of L(G) arranged in nonincreasing order. Especially, ℓ (G)is called Laplacian spectral

.Definition ..

...

Let G = (V, E) be a simple connected graph with vertex set V ={v1, v2,· · · , vn} and edge set E.Then

.

1.. v_i ∼ vj means that v_i and v_j are adjacent in G.

. ..

2 mi = 1

di

∑

vj∼vⁱdj is called average 2-degree of vertex v_i. .

3.. A complement of G is a graph with the same vertices as G has and with those and only those edges which do not appear in G.

The graph is denoted by G^c.

In 1985, Anderson and Morley showed the following bound ℓ1(G)≤ max

vi∼vj{di+ dj} . (1)

In 1998, Merris improved the bound (1), as follows ℓ1(G)≤ max

vi∈V (G){di+ mi} . (2)

In 2000, Rojo et al. showed the following upper bound ℓ1(G)≤ max

vi∼vj{di+ dj− |G1(vi)∩ G1(vj)|} . (3) In 2001, Li and Pan gave a bound, as follows

ℓ₁(G)≤ max

vi∈V (G)

{√2d_i(d_i+ m_i) }

. (4)

In 2004, Zhang showed the following result, which is always better than the bound (4).

Outline

1... Introdution

2... Preliminaries

3... Main Results

Some Corollary about Theorem 1 Main Theorem

Applications of the main theorem

4... Conjecture

We have the following facts about L(G) and S(G).

. ..

1 L(G)is positive semi-definite.

.

2.. ℓn(G) = 0is an eigenvalue of L(G) corresponding to the eigenvector 1_n, where 1_nis the all-ones vector.

. ..

3 If X = (x₁, x2,· · · , xn)^⊤is an eigenvector of L(G) corresponding to ℓ_i(G) (1≤ i ≤ n − 1), then∑_n

i=1x_i= 0.

.

4.. L(G) + L(G^c) = nI − J, where I and J are identity matrix and all-ones matrix, respectively.

.

5.. If X is the eigenvector of L(G) corresponding to ℓ_i(G) (1≤ i ≤ n − 1), then X is also an eigenvector of L(G^c) corresponding to n− ℓi(G).

.

6.. ℓ_i(G)≤ n, for 1 ≤ i ≤ n.

.Definition ..

...

Let G = (V, E) be a simple connected graph with vertex set V ={v1, v2,· · · , vn} and edge set E. The following notations are adopted.

. ..

1 λ(G) = min

vi∼vj|G1(vi)∩ G1(vj)|.

. ..

2 µ(G) = min

vivj|G1(vi)∩ G1(vj)|.

In 2013, Guo et al. improve the bound (5) and showed the following theorem.

.Theorem 1 ..

...

Let G = (V, E) be a simple connected graph with vertex set V ={v1, v2,· · · , vn} and edge set E. We define

M (G) = max

vi∈V (G)

{2d_i− λ +√

4d_im_i− 4λdi+ λ² 2

} .

Then

ℓ1(G)≤ M(G), (6)

where λ = λ(G).

Outline

1... Introdution

2... Preliminaries

3... Main Results

Some Corollary about Theorem 1 Main Theorem

Applications of the main theorem

4... Conjecture

Outline

1... Introdution

2... Preliminaries

3... Main Results

Some Corollary about Theorem 1 Main Theorem

Applications of the main theorem

4... Conjecture

We have two corollaries about Theorem 1.

.Corollary 3 ..

...

If G is a k-regular graph, then

ℓ₁(G)6 2k − λ, where λ = λ(G).

.Corollary 4 ..

...

If G is a simple connected graph with n vertices, then ℓ₁(G)≤ min {M(G), n} .

Outline

1... Introdution

2... Preliminaries

3... Main Results

Some Corollary about Theorem 1 Main Theorem

Applications of the main theorem

4... Conjecture

.Proposition 5 ..

...

Let G = (V, E) be a simple connected graph with vertex set V ={v1, v2,· · · , vn} and edge set E.

. ..

1 If T = A(G)² and T = (t_ij), we have tij =|G1(vi)∩ G1(vj)| and

∑n j=1

tij = ∑

vj∼vi

dj = midi.

. ..

2 If X = (x₁, x2,· · · , xn)^⊤is a vector, X^⊤L(G)X = ∑

j<k vj∼vk

(xj − xk)².

.Theorem 6

..Let G = (V, E) be a simple connected graph with vertex set V ={v1, v₂,· · · , vn} and edge set E. Let

S(G) = (ℓ1(G), ℓ2(G),· · · , ℓn(G))be the Laplacian spectrum of G. We define

M^′(G) = max

vi∈V (G)

{2di− λ + µ +√ Bi

2 : Bi ≥ 0

}

and

N^′(G) = min

vi∈V (G)

{2di− λ + µ −√ Bi

2 : Bi ≥ 0

} ,

where B_i= 4dimi− 4(λ − µ)di+ (λ− µ)²− 4µn, λ = λ(G), and µ = µ(G). Then

N^′(G)≤ ℓ(G) ≤ M^′(G), (7)

.proof(cont.) ..

Let X = (x₁, x2,· · · , xn)^⊤be the eigenvector of L(G) corresponding to ℓ(G). We have

∑n

i=1

[di− ℓ(G)]²x²_i =∥(D(G) − ℓ(G)I)X∥²

=∥(D(G) − L(G))X∥²

=∥A(G)X∥²

=X^⊤T X

=

∑n

i=1

tiix²_i+ 2∑

j<k

tjkxjxk

=

∑n

i=1

tiix²_i+∑

j<k

tjk(x²_j+ x²_k− (xj− xk)²)

=

∑n

i=1

((tii+

∑n

j=1j̸=i

tij)x²i)− ∑

j<k v∼v

tjk(xj− xk)²− ∑

j<k v v

tjk(xj− xk)²

.Proof.

..

∑n

i=1

[di− ℓ(G)]²x²i =

∑n

i=1

((tii+

∑n

j=1 j̸=i

tij)x²i)− ∑

j<k v_j∼vk

tjk(xj− xk)²− ∑

j<k v_jvk

tjk(xj− xk)²

≤

∑n

i=1

dimix²i− λ ∑

j<k v_j∼vk

(xj− xk)²− µ ∑

j<k v_jvk

(xj− xk)²

=

∑n

i=1

dimix²i− λX^⊤L(G)X− µX^⊤L(G^c)X

=

∑n

i=1

dimix²i− λℓ(G)∥X∥²− µ(n − ℓ(G))∥X∥²

=

∑n

i=1

dimix²_i− λℓ(G)

∑n

i=1

x²_i− µ(n − ℓ(G))

∑n

i=1

x²_i.

.proof(cont.) ..Thus, we have

∑n i=1

[(di− ℓ(G))²− dimi+ λℓ(G) + µ(n− ℓ(G))]x²i ≤ 0. (8)

Then there must exist a vertex visuch that

(di− ℓ(G))²− dimi+ λℓ(G) + µ(n− ℓ(G))

=ℓ(G)²− (2di− λ + µ)ℓ(G) + (d²i − dim_i+ µn)≤ 0, which implies that

2d_i− λ + µ −√ B_i

2 ≤ ℓ(G) ≤ 2d_i− λ + µ +√ B_i

2 .

Therefore,

N^′(G)≤ ℓ(G) ≤ M^′(G).

When ℓ(G) = ℓ₁(G)or ℓ(G) = ℓ_n−1(G), we have the following inequalities about ℓ₁(G)and ℓ_n−1(G).

.Theorem 7 ..

...

Let G be a simple connected graph. Then

ℓ₁(G)≤ M^′(G) (9)

and

ℓ_n−1(G)≥ N^′(G) (10)

.Definition ..

...

We call G is a strongly regular graph with parameter (n, k, λ, µ), if G is a k-regular graph with n vertices and common neighbours of two adjacent/nonadjcent vertices is a fixed number λ/µ, respectively, where µ̸= 0 and G is denoted by srg(n, k, λ, µ).

.Remark ..

...

n = 1 + k + k(k− 1 − λ)

µ .

.Example 1

..In this example, G is the Petersen graph which is srg(10, 3, 0, 1), as follows.

..

. .

3

.

4

. 5

.

6

.

7

.

8

.

9

.

10

.Example 1(cont.) ..

We have λ = 0, µ = 1, and d_i = 3, for any vertex v_i, and we compute ℓ₁(G) = 5. We calculate

M (G) = max

v_i∈V (G)

{2di− λ +√

4dimi− 4λdi+ λ² 2

}

=2× 3 − 0 +√

4× 3²− 0 + 0 2

=6.

M^′(G) = max

vi∈V (G)

{

2di− λ + µ +√

4dimi− 4(λ − µ)di+ (λ− µ)²− 4µn 2

}

=2× 3 − 0 + 1 +√

4× 3²− 4(0 − 1)3 + (0 − 1)²− 4 × 1 × 10 2

=5.

′ ≤ M(G) = 6.

.Corollay 8 ..

...

If G is a simple connected graph with n vertices, then ℓ₁(G)≤ min{

M^′(G), n} .

.Theorem 9 ..

...

Let G = (V, E) be a simple connected graph with vertex set V ={v1, v₂,· · · , vn} and edge set E. Then

min{

M^′(G), n}

≤ min {M(G), n} .

.Sketch the proof of Theorem 8 ..

...

Case 1: When M (G)≥ n,

we have min{M(G), n} = n ≥ min {M^′(G), n}

Case 2: When M (G) < n. Let ζ_i be the largest root of

fi(x) = (di− x)²− dimi+ λx = 0and ξ_i be the largest root of g_i(x) = (d_i− x)²− dim_i+ λx + µ(n− x) = 0, for 1 ≤ i ≤ n, where λ = λ(G)and µ = µ(G). Then we have

ζ_i= 2di− λ +√

4dimi− 4λdi+ λ² 2

and

ξi= 2di− λ + µ +√

4dimi− 4(λ − µ)di+ (λ− µ)²− 4µn

2 .

.Sketch the proof of Theorem 8(cont.) ..

..

ξi

.

ζi

.

(ζi, gi(ζi))

.

y = g_i(x)

Outline

1... Introdution

2... Preliminaries

3... Main Results

Some Corollary about Theorem 1 Main Theorem

Applications of the main theorem

4... Conjecture

In 1998, R. Merris got the following result.

.Definition ..

...

Let G₁= (V₁, E₁)and G₂= (V₂, E₂)be two graphs with disjoint vertex sets. Then we define the join of two graphs G₁and G₂ is

G₁∨ G2 = (V, E), where V = V₁∪ V2 and E = E₁∪ E2∪ {xy|x ∈ V1and y∈ V2} . .Theorem

..

Let G₁= (V₁, E₁)and G₂= (V₂, E₂)be two graphs with disjoint vertex sets and (|V1|, |V2|) = (n, m). Let λi and ν_j be eigenvalue of L(G₁)and L(G₂)corresponding to the eigenvector v_iand w_j, respectively, where

< λ_i >and < ν_j >both are nonincreasing sequences, for all 1≤ i ≤ n and 1≤ j ≤ m. Then, 0, λi+ m, νj + n, and n + m are eigenvalues of L(G₁∨ G2)corresponding to the eigenvector 1_n+m, (v_i^⊤, 0^⊤_m)^⊤,

⊤ ⊤ ⊤ ⊤ −n1^{⊤ ⊤} ≤ i ≤ n and

.Corollary 10 ..

...

If G is k-regular graph, then

ℓ1(G)≤2k− λ + µ +√

4k²− 4(λ − µ)k + (λ − µ)²− 4µn 2

and

ℓ_n−1(G)≥ 2k− λ + µ −√

4k²− 4(λ − µ)k + (λ − µ)²− 4µn

2 .

.Corollary 11 ..

...

If G is a strongly regular graph with parameters (n, k, λ, µ), then

ℓ1(G) = M^′(G) = 2k− λ + µ +√

(λ− µ)²+ 4(k− µ) 2

and

ℓ_n−1(G) = N^′(G) = 2k− λ + µ −√

(λ− µ)²+ 4(k− µ)

2 .

.Example 2

..We usually call F_ℓ = K1∨ ℓK2be a fan graph.

. .

1 .

4

. 2

. 3 .

5

When G = F₂, we have

A(G) =









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







, L(G) =









2 0 0 −1 −1

0 2 −1 0 −1

0 −1 2 0 −1

−1 0 0 2 −1

−1 −1 −1 −1 4







.

.Example 2(cont.) ..

Hence, λ = 1, µ = 1, and X =







 1 1 1 1

−4







is a eigenvector corresponding to the eigenvalue ℓ₁(G) = 5.

We calculate M^′(G)and the equality in (8) as shown in the following table.

i d_i m_i ξ_i ϕ_i

1∼ 4 2 3 3 (2− 5)²− 2 · 3 + 1 · 5 + 1 · (5 − 5) = 8 5 4 2 ^8+√₂¹² ≈ 5.73 (4 − 5)²− 4 · 2 + 1 · 5 + 1 · (5 − 5) = −2 ℓ1(G) = 5 < ⁸⁺

√12

2 ,so the inequality (9) does not hold. But the

∑5 ₂

Example 3/Example 4 are some graphs, which satisfy the equality in (8) with n = 5/n = 6.

.Example 3 ..

...

..

1 .

2 .

3

. 4

. 5

..

1 .

2 .

3

. 4

. 5

..

1 .

2 .

3

. 4

. 5

K1,4 K2,3 K5

.Example 3(cont.) ..

G L(G) M^′(G) ℓ1(G)

K_1,4









4 −1 −1 −1 −1

−1 1 0 0 0

−1 0 1 0 0

−1 0 0 1 0

−1 0 0 0 1









8 +√ 16

2 = 6 5

K2,3









3 0 −1 −1 −1 0 3 −1 −1 −1

−1 −1 2 0 0

−1 −1 0 2 0

−1 −1 0 0 2









8 +√ 12

2 ≈ 5.73 5

K







4 −1 −1 −1 −1

−1 4 −1 −1 −1

−1 −1 4 −1 −1





 5 5

.Example 4 ..

..

1 .

2 .

3

.

4

. 5

. 6

..

1 .

2 .

3

.

4

. 5

. 6

..

1 .

2 .

3

.

4

. 5

. 6

C₆ K_1,5 K_2,4

ℓ1(G) = M^′(G) ℓ1(G)̸= M^′(G) ℓ1(G)̸= M^′(G)

..

1 .

2 .

3

.

4

. 5

. 6

..

1 .

2 .

3

.

4

. 5

. 6

..

1 .

2 .

3

.

4

. 5

. 6

K_3,3 K_2,2,2 K₆

.Corollary 12 ..

...

Let G be a complete k-partite graph(k≥ 2). Then, ℓ1(G) = M^′(G)if and only if every part in G has the same vertices.

.Example 6

..In this example, we have a graph, which are not k-partite graph or strongly regular graph. We have λ = 0, µ = 1, di= 3, for all vertex vi. Then

M^′(G) = 6− 0 + 1 +√

4× 9 − 4(−1)3 + (−1)²− 4(1)(8)

2 =7 +√

17

2 = ℓ₁(G)

..

. .

3 .

4

.

5

.

6

. 7

. 8

Outline

1... Introdution

2... Preliminaries

3... Main Results

Some Corollary about Theorem 1 Main Theorem

Applications of the main theorem

4... Conjecture

.Conjecture ..

...

Let G be a simple connected graph. If G satisfy M^′(G) = ℓ1(G), then Gis a regular graph.

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J.S. Li, Y.L. Pan, De Caen’s inequality and bounds on the largest Laplacian eigenvalue of a graph,

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R. Merris, Laplacian graph eigenvectors,

Linear Algebra and its Applications 278 (1998), 221-236.

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Linear Algebra and its Application 376 (2004), 207-213.

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