The Laplacian spectral radius of a graph

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The Laplacian spectral radius of a graph

Fan-Hsuan Lin Advisor: Chih-Wen Weng

National Chiao Tung University

August 2, 2014

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Outline

1... Introdution

2... Preliminaries

3... Main Results

Some Corollary about Theorem 1 Main Theorem

Applications of the main theorem

4... Conjecture

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Outline

1... Introdution

2... Preliminaries

3... Main Results

Some Corollary about Theorem 1 Main Theorem

Applications of the main theorem

4... Conjecture

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.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 di=|G1(vi)| the degree of vertex vi ∈ V (G), where G1(vi)is the set of neighbors of vi, 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

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.Definition ..

...

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

.

1.. vi ∼ vj means that vi and vj are adjacent in G.

. ..

2 mi = 1

di

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

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 Gc.

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In 1985, Anderson and Morley showed the following bound 1(G)≤ max

vi∼vj{di+ dj} . (1)

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

vi∈V (G){di+ mi} . (2)

In 2000, Rojo et al. showed the following upper bound 1(G)≤ max

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

1(G)≤ max

vi∈V (G)

{√2di(di+ mi) }

. (4)

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

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Outline

1... Introdution

2... Preliminaries

3... Main Results

Some Corollary about Theorem 1 Main Theorem

Applications of the main theorem

4... Conjecture

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

. ..

1 L(G)is positive semi-definite.

.

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

. ..

3 If X = (x1, x2,· · · , xn)is an eigenvector of L(G) corresponding to ℓi(G) (1≤ i ≤ n − 1), thenn

i=1xi= 0.

.

4.. L(G) + L(Gc) = 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 ℓi(G) (1≤ i ≤ n − 1), then X is also an eigenvector of L(Gc) corresponding to n− ℓi(G).

.

6.. i(G)≤ n, for 1 ≤ i ≤ n.

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.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)|.

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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)

{2di− λ +√

4dimi− 4λdi+ λ2 2

} .

Then

1(G)≤ M(G), (6)

where λ = λ(G).

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Outline

1... Introdution

2... Preliminaries

3... Main Results

Some Corollary about Theorem 1 Main Theorem

Applications of the main theorem

4... Conjecture

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Outline

1... Introdution

2... Preliminaries

3... Main Results

Some Corollary about Theorem 1 Main Theorem

Applications of the main theorem

4... Conjecture

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We have two corollaries about Theorem 1.

.Corollary 3 ..

...

If G is a k-regular graph, then

1(G)6 2k − λ, where λ = λ(G).

.Corollary 4 ..

...

If G is a simple connected graph with n vertices, then 1(G)≤ min {M(G), n} .

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Outline

1... Introdution

2... Preliminaries

3... Main Results

Some Corollary about Theorem 1 Main Theorem

Applications of the main theorem

4... Conjecture

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.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 = (tij), 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, XL(G)X =

j<k vj∼vk

(xj − xk)2.

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

N(G)≤ ℓ(G) ≤ M(G), (7)

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.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)]2x2i =∥(D(G) − ℓ(G)I)X∥2

=∥(D(G) − L(G))X∥2

=∥A(G)X∥2

=XT X

=

n

i=1

tiix2i+ 2

j<k

tjkxjxk

=

n

i=1

tiix2i+

j<k

tjk(x2j+ x2k− (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

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.Proof.

..

n

i=1

[di− ℓ(G)]2x2i =

n

i=1

((tii+

n

j=1 j̸=i

tij)x2i)

j<k vj∼vk

tjk(xj− xk)2

j<k vjvk

tjk(xj− xk)2

n

i=1

dimix2i− λ

j<k vj∼vk

(xj− xk)2− µ

j<k vjvk

(xj− xk)2

=

n

i=1

dimix2i− λXL(G)X− µXL(Gc)X

=

n

i=1

dimix2i− λℓ(G)∥X∥2− µ(n − ℓ(G))∥X∥2

=

n

i=1

dimix2i− λℓ(G)

n

i=1

x2i− µ(n − ℓ(G))

n

i=1

x2i.

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.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 − dimi+ µn)≤ 0, which implies that

2di− λ + µ − Bi

2 ≤ ℓ(G) ≤ 2di− λ + µ + Bi

2 .

Therefore,

N(G)≤ ℓ(G) ≤ M(G).

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

.Theorem 7 ..

...

Let G be a simple connected graph. Then

1(G)≤ M(G) (9)

and

n−1(G)≥ N(G) (10)

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.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 .

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.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

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.Example 1(cont.) ..

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

M (G) = max

vi∈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.

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.Corollay 8 ..

...

If G is a simple connected graph with n vertices, then 1(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} .

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.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 ζi be the largest root of

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

ζi= 2di− λ +√

4dimi− 4λdi+ λ2 2

and

ξi= 2di− λ + µ +

4dimi− 4(λ − µ)di+ (λ− µ)2− 4µn

2 .

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

..

ξi

.

ζi

.

i, gii))

.

y = gi(x)

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Outline

1... Introdution

2... Preliminaries

3... Main Results

Some Corollary about Theorem 1 Main Theorem

Applications of the main theorem

4... Conjecture

(28)

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 νj be eigenvalue of L(G1)and L(G2)corresponding to the eigenvector viand wj, respectively, where

< λi >and < νj >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 1n+m, (vi, 0m),

−n1 ≤ i ≤ n and

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.Corollary 10 ..

...

If G is k-regular graph, then

1(G)2k− λ + µ +

4k2− 4(λ − µ)k + (λ − µ)2− 4µn 2

and

n−1(G) 2k− λ + µ −

4k2− 4(λ − µ)k + (λ − µ)2− 4µn

2 .

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.Corollary 11 ..

...

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

1(G) = M(G) = 2k− λ + µ +

(λ− µ)2+ 4(k− µ) 2

and

n−1(G) = N(G) = 2k− λ + µ −

(λ− µ)2+ 4(k− µ)

2 .

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.Example 2

..We usually call F = 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





.

(32)

.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 di mi ξi ϕi

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 1(G) = 5 < 8+

12

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

5 2

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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

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.Example 3(cont.) ..

G L(G) M(G) 1(G)

K1,4

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

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.Example 4 ..

..

1 .

2 .

3

.

4

. 5

. 6

..

1 .

2 .

3

.

4

. 5

. 6

..

1 .

2 .

3

.

4

. 5

. 6

C6 K1,5 K2,4

1(G) = M(G) 1(G)̸= M(G) 1(G)̸= M(G)

..

1 .

2 .

3

.

4

. 5

. 6

..

1 .

2 .

3

.

4

. 5

. 6

..

1 .

2 .

3

.

4

. 5

. 6

K3,3 K2,2,2 K6

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.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.

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.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

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Outline

1... Introdution

2... Preliminaries

3... Main Results

Some Corollary about Theorem 1 Main Theorem

Applications of the main theorem

4... Conjecture

(39)

.Conjecture ..

...

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

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W.N. Anderson, T.D. Morley, Eigenvalues of the Laplacian of a graph,

Linear Multilinear Algebra, 18 (1985), 141-145.

A.E. Brouwer and W. H. Haemers Spectra of Graphs, 2012: Springer-Verlag

Miroslav Fiedler, Praha, Algebraic connectivity of graphs, Czechoslovak Mathematical Journal 23 (98) 1973, Praha.

J.M. Guo, J. Li, W.C. Shiu, A note on the upper bounds for the Laplacian spectral radius of graphs,

(41)

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

Linear Algebra and its Applications, 328 (2001), 153-160.

R. Merris, Laplacian graph eigenvectors,

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

R. Merris, A note on Laplacian graph eigenvalues, Linear Algebra and its Applications, 285 (1998), 33-35.

M. W. Newman, The Laplacian Spectrum of Graphs , University of Manitoba, Winnipeg, MB, Canada, 2000.

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O. Rojo, R. Soto, H. Rojo, , An always nontrivial upper bound for Laplacian graph eigenvalues,

Linear Algebra and its Applications, 312 (2000), 155-159.

L.S. Shi, Bounds on the (Laplacian) spectral radius of graphs, Linear Algebra and its Applications, 422 (2007), 755-770.

X.D. Zhang, Two sharp upper bounds for the Laplacian eigenvalues,

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

X.D. Zhang, R. Luo, The Laplacian eigenvalues of mixed graphs,

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