### 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 ={v*1*, v*2*,· · · , v**n***} and edge set E. Let A(G) be the adjacency****matrix of G. Denote by d*** _{i}*=

*|G*1

*(v*

*)*

_{i}*| the degree of vertex v*

*i*

*∈ V (G),*

*where G*

_{1}

*(v*

*)*

_{i}*is the set of neighbors of v*

*, and let*

_{i}*D(G)=diag(d*_{1}*, d*2*,· · · , d**n*) be the diagonal matrix with entries
*d*_{1}*, d*_{2}*,· · · , d**n*. 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) = (ℓ*_{1}*(G), ℓ*_{2}*(G),· · · , ℓ**n**(G)),*

*where ℓ*_{1}*(G)≥ ℓ*2*(G)≥ · · · ≥ ℓ**n**(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 ={v*1*, v*2*,· · · , v**n**} and edge set E.Then*

.

1.. *v*_{i}*∼ v**j* *means that v*_{i}*and v*_{j}*are adjacent in G.*

. ..

2 *m**i* = 1

*d**i*

∑

*v**j**∼v*^{i}*d**j* **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
*ℓ*1*(G)≤ max*

*v**i**∼v**j**{d**i**+ d**j**} .* (1)

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

*v**i**∈V (G)**{d**i**+ m**i**} .* (2)

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

*v**i**∼v**j**{d**i**+ d**j**− |G*1*(v**i*)*∩ G*1*(v**j*)|} . (3)
In 2001, Li and Pan gave a bound, as follows

*ℓ*_{1}*(G)≤ max*

*v**i**∈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.. *ℓ**n**(G) = 0is an eigenvalue of L(G) corresponding to the*
**eigenvector 1**_{n}**, where 1*** _{n}*is the all-ones vector.

. ..

3 *If X = (x*_{1}*, x*2*,· · · , x**n*)^{⊤}*is an eigenvector of L(G) corresponding*
*to ℓ*_{i}*(G) (1≤ i ≤ n − 1), then*∑_{n}

*i=1**x** _{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− ℓ*

*i*

*(G).*

.

6.. *ℓ*_{i}*(G)≤ n, for 1 ≤ i ≤ n.*

.Definition ..

...

*Let G = (V, E) be a simple connected graph with vertex set*
*V ={v*1*, v*2*,· · · , v**n**} and edge set E. The following notations are*
adopted.

. ..

1 *λ(G) = min*

*v**i**∼v**j**|G*1*(v**i*)*∩ G*1*(v**j*)|.

. ..

2 *µ(G) = min*

*v**i**v**j**|G*1*(v**i*)*∩ G*1*(v**j*)*|.*

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 ={v*1*, v*2*,· · · , v**n**} and edge set E. We define*

*M (G) = max*

*v**i**∈V (G)*

{*2d*_{i}*− λ +√*

*4d*_{i}*m*_{i}*− 4λd**i**+ λ*^{2}
2

}
*.*

Then

*ℓ*1*(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*

*ℓ*_{1}*(G)6 2k − λ,*
*where λ = λ(G).*

.Corollary 4 ..

...

*If G is a simple connected graph with n vertices, then*
*ℓ*_{1}*(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 ={v*1*, v*2*,· · · , v**n**} and edge set E.*

. ..

1 *If T = A(G)*^{2} *and T = (t** _{ij}*), we have

*t*

*ij*=

*|G*1

*(v*

*i*)

*∩ G*1

*(v*

*j*)| and

∑*n*
*j=1*

*t**ij* = ∑

*v**j**∼v**i*

*d**j* *= m**i**d**i**.*

. ..

2 *If X = (x*_{1}*, x*2*,· · · , x**n*)^{⊤}*is a vector, X*^{⊤}*L(G)X =* ∑

*j<k*
*v**j**∼v**k*

*(x**j* *− x**k*)^{2}*.*

.Theorem 6

..*Let G = (V, E) be a simple connected graph with vertex set*
*V ={v*1*, v*_{2}*,· · · , v**n**} and edge set E. Let*

*S(G) = (ℓ*1*(G), ℓ*2*(G),· · · , ℓ**n**(G))be the Laplacian spectrum of G. We*
define

*M*^{′}*(G) = max*

*v**i**∈V (G)*

{*2d**i**− λ + µ +√*
*B**i*

2 *: B**i* *≥ 0*

}

and

*N*^{′}*(G) =* min

*v**i**∈V (G)*

{*2d**i**− λ + µ −√*
*B**i*

2 *: B**i* *≥ 0*

}
*,*

*where B*_{i}*= 4d**i**m**i**− 4(λ − µ)d**i**+ (λ− µ)*^{2}*− 4µn,*
*λ = λ(G),* *and µ = µ(G). Then*

*N*^{′}*(G)≤ ℓ(G) ≤ M*^{′}*(G),* (7)

.proof(cont.) ..

*Let X = (x*_{1}*, x*2*,· · · , x**n*)^{⊤}*be the eigenvector of L(G) corresponding to*
*ℓ(G). We have*

∑*n*

*i=1*

*[d**i**− ℓ(G)]*^{2}*x*^{2}* _{i}* =

*∥(D(G) − ℓ(G)I)X∥*

^{2}

=*∥(D(G) − L(G))X∥*^{2}

=∥A(G)X∥^{2}

*=X*^{⊤}*T X*

=

∑*n*

*i=1*

*t**ii**x*^{2}* _{i}*+ 2∑

*j<k*

*t**jk**x**j**x**k*

=

∑*n*

*i=1*

*t**ii**x*^{2}* _{i}*+∑

*j<k*

*t**jk**(x*^{2}_{j}*+ x*^{2}_{k}*− (x**j**− x**k*)^{2})

=

∑*n*

*i=1*

*((t**ii*+

∑*n*

*j=1**j̸=i*

*t**ij**)x*^{2}*i*)*−* ∑

*j<k*
*v**∼v*

*t**jk**(x**j**− x**k*)^{2}*−* ∑

*j<k*
*v* *v*

*t**jk**(x**j**− x**k*)^{2}

.Proof.

..

∑*n*

*i=1*

*[d**i**− ℓ(G)]*^{2}*x*^{2}*i* =

∑*n*

*i=1*

*((t**ii*+

∑*n*

*j=1*
*j**̸=i*

*t**ij**)x*^{2}*i*)*−* ∑

*j<k*
*v*_{j}*∼v**k*

*t**jk**(x**j**− x**k*)^{2}*−* ∑

*j<k*
*v*_{j}*v**k*

*t**jk**(x**j**− x**k*)^{2}

*≤*

∑*n*

*i=1*

*d**i**m**i**x*^{2}*i**− λ* ∑

*j<k*
*v*_{j}*∼v**k*

*(x**j**− x**k*)^{2}*− µ* ∑

*j<k*
*v*_{j}*v**k*

*(x**j**− x**k*)^{2}

=

∑*n*

*i=1*

*d**i**m**i**x*^{2}*i**− λX*^{⊤}*L(G)X**− µX*^{⊤}*L(G*^{c}*)X*

=

∑*n*

*i=1*

*d**i**m**i**x*^{2}*i**− λℓ(G)∥X∥*^{2}*− µ(n − ℓ(G))∥X∥*^{2}

=

∑*n*

*i=1*

*d**i**m**i**x*^{2}_{i}*− λℓ(G)*

∑*n*

*i=1*

*x*^{2}_{i}*− µ(n − ℓ(G))*

∑*n*

*i=1*

*x*^{2}_{i}*.*

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

∑*n*
*i=1*

*[(d**i**− ℓ(G))*^{2}*− d**i**m**i**+ λℓ(G) + µ(n**− ℓ(G))]x*^{2}*i* *≤ 0.* (8)

*Then there must exist a vertex v**i*such that

*(d**i**− ℓ(G))*^{2}*− d**i**m**i**+ λℓ(G) + µ(n**− ℓ(G))*

*=ℓ(G)*^{2}*− (2d**i**− λ + µ)ℓ(G) + (d*^{2}*i* *− d**i**m*_{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) = ℓ*_{1}*(G)or ℓ(G) = ℓ*_{n}_{−1}*(G), we have the following*
*inequalities about ℓ*_{1}*(G)and ℓ*_{n}_{−1}*(G).*

.Theorem 7 ..

...

*Let G be a simple connected graph. Then*

*ℓ*_{1}*(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

*, and we compute*

_{i}*ℓ*

_{1}

*(G) = 5. We calculate*

*M (G) = max*

*v*_{i}*∈V (G)*

{*2d**i**− λ +**√*

*4d**i**m**i**− 4λd**i**+ λ*^{2}
2

}

=2*× 3 − 0 +**√*

4*× 3*^{2}*− 0 + 0*
2

*=6.*

*M*^{′}*(G) = max*

*v**i**∈V (G)*

{

*2d**i**− λ + µ +*√

*4d**i**m**i**− 4(λ − µ)d**i**+ (λ**− µ)*^{2}*− 4µn*
2

}

=2*× 3 − 0 + 1 +*√

4*× 3*^{2}*− 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*
*ℓ*_{1}*(G)≤ min*{

*M*^{′}*(G), n*}
*.*

.Theorem 9 ..

...

*Let G = (V, E) be a simple connected graph with vertex set*
*V ={v*1*, v*_{2}*,· · · , v**n**} 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

*f**i**(x) = (d**i**− x)*^{2}*− d**i**m**i**+ λx = 0and ξ** _{i}* be the largest root of

*g*

_{i}*(x) = (d*

_{i}*− x)*

^{2}

*− d*

*i*

*m*

_{i}*+ λx + µ(n− x) = 0, for 1 ≤ i ≤ n, where*

*λ = λ(G)and µ = µ(G). Then we have*

*ζ** _{i}*=

*2d*

*i*

*− λ +√*

*4d**i**m**i**− 4λd**i**+ λ*^{2}
2

and

*ξ**i*= *2d**i**− λ + µ +*√

*4d**i**m**i**− 4(λ − µ)d**i**+ (λ− µ)*^{2}*− 4µn*

2 *.*

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

..

*ξ**i*

.

*ζ**i*

.

*(ζ**i**, g**i**(ζ**i*))

.

*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 G*_{1}*= (V*_{1}*, E*_{1})*and G*_{2}*= (V*_{2}*, E*_{2})be two graphs with disjoint vertex
**sets. Then we define the join of two graphs G**_{1}*and G*_{2} is

*G*_{1}*∨ G*2 *= (V, E), where V = V*_{1}*∪ V*2 and
*E = E*_{1}*∪ E*2*∪ {xy|x ∈ V*1*and y∈ V*2*} .*
.Theorem

..

*Let G*_{1}*= (V*_{1}*, E*_{1})*and G*_{2}*= (V*_{2}*, E*_{2})be two graphs with disjoint vertex
sets and (*|V*1*|, |V*2*|) = (n, m). Let λ**i* *and ν*_{j}*be eigenvalue of L(G*_{1})and
*L(G*_{2})*corresponding to the eigenvector v*_{i}*and w** _{j}*, respectively, where

*< λ*_{i}*>and < ν*_{j}*>*both are nonincreasing sequences, for all 1*≤ i ≤ n*
and 1*≤ j ≤ m. Then, 0, λ**i**+ m, ν**j* *+ n, and n + m are eigenvalues of*
*L(G*_{1}*∨ G*2)**corresponding to the eigenvector 1**_{n+m}*, (v*_{i}^{⊤}**, 0**^{⊤}* _{m}*)

*,*

^{⊤}*⊤* *⊤* *⊤* *⊤* **−n1**^{⊤}^{⊤}*≤ i ≤ n and*

.Corollary 10 ..

...

*If G is k-regular graph, then*

*ℓ*1*(G)**≤**2k**− λ + µ +*√

*4k*^{2}*− 4(λ − µ)k + (λ − µ)*^{2}*− 4µn*
2

and

*ℓ*_{n}_{−1}*(G)**≥* *2k**− λ + µ −*√

*4k*^{2}*− 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*

*ℓ*1*(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}*= K*1*∨ ℓK*2be a fan graph.

. .

1 .

4

. 2

. 3 .

5

*When G = F*_{2}, 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 ℓ*_{1}*(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+}^{√}_{2}^{12} *≈ 5.73 (4 − 5)*^{2}*− 4 · 2 + 1 · 5 + 1 · (5 − 5) = −2*
*ℓ*1*(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

*K**1,4* *K**2,3* *K*5

.Example 3(cont.) ..

*G* *L(G)* *M*^{′}*(G)* *ℓ*1*(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

*K**2,3*

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

*C*_{6} *K*_{1,5}*K*_{2,4}

*ℓ*1*(G) = M*^{′}*(G)* *ℓ*1*(G)̸= M*^{′}*(G)* *ℓ*1*(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}*K*_{6}

.Corollary 12 ..

...

*Let G be a complete k-partite graph(k≥ 2). Then, ℓ*1*(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, d**i*= 3, for all vertex v*i*. Then

*M*^{′}*(G) =* 6*− 0 + 1 +*√

4*× 9 − 4(−1)3 + (−1)*^{2}*− 4(1)(8)*

2 =7 +*√*

17

2 *= ℓ*_{1}*(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) = ℓ*1*(G), then*
*G*is a regular graph.

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,

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.

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,