2012/8/19,  上海, A few inequalities related to spectral excess theorem,

.

...

A few inequalities related to spectral excess theorem

Chih-wen Weng (翁志文)

(joint work with Guang-Siang Lee(李光祥))

Department of Applied Mathematics National Chiao Tung University

August 19, 2012

f (eigenvalues) ≤ g

(graph parameters eigenvectors

)

≤ h(eigenvalues).

quasi-spectral characterization

?

quasi-spectral characterization

?

spectral characterization

 I

(still working)

Preliminaries

.

1

..

Throughout let G = (V G, EG) be a simple connected graph of order n and diameter D.

.

2

..

Assume that adjacency matrix A has d + 1 distinct eigenvalues λ

0 >

λ

1 > . . . >

λ

d

with corresponding multiplicities

1 = m

₀ , m ₁ , ··· ,m d

. d is called the

spectral diameter

of G.

.

3

..

It is well-known that D

≤ d.

.

4

..

Z(x) :=

∏ d i=0

(x

−

λ

i

) is the

minimal polynomial

of A.

Inner product

Consider the vector spaceR

d

[x] ∼=R[x]/⟨Z(x)⟩ with the inner product

⟨p(x),q(x)⟩ :=

1

n

tr(p(A)q(A)) =1

n ∑

i, j

(p(A)

◦ q(A)) i j ,

for p(x), q(x)

∈ R d

[x], where

◦ is the entrywise product of matrices.

Predistance polynomials

.

Definition 1.1

..

...

(i) The orthogonal polynomials 1 = p

0

(x), p

1

(x), . . . , p

d

(x) in R

d

[x] satisfying

deg p

i

(x) = i and

⟨p i

(x), p

_j

(x)

⟩ =

δ

i j p i

(λ

0

) are called the

predistance polynomials

of G.

(ii) The polynomial

H(x) := n

∏ d i=1

x −

λ

i

λ

0 −

λ

i

is called the

Hoffman polynomial

of G. Moreover, G is regular iff H(A) = J, the all 1’s matrix.

The sum of all predistance polynomials gives the Hoffman polynomial

H(x) = p ₀

(x) + p

₁

(x) +

··· + p d

(x) and

H(A) = p ₀

(A) + p

₁

(A) +

··· + p d

(A).

Three -term recurrence

The predistance polynomials satisfy a three-term recurrence:

xp _i

(x) = c

^′ _i+1 p _i+1

(x) + a

^′ _i p _i

(x) + b

^′ _{i −1} p _i−1

(x) 0

≤ i ≤ d,

where c

^′ _i+1 , a ^′ _i , b ^′ _i−1 ∈ R with b ^′ ₋₁

= c

^′ _d+1

:= 0.

Three -term recurrence for bipartite graph

.

1

..

If G is bipartite, then the predistance polynomials satisfy a three-term recurrence of the form

x ² p i

(x) = X

_i+2 p i+2

(x) +Y

_i p i

(x) + Z

_{i −2} p i −2

(x) 0

≤ i ≤ d,

(1) where

X i+2

= c

^′ _i+1 c ^′ _i+2 , Y _i

= b

^′ _i c ^′ _i+1

+ b

^′ _i−1 c ^′ _i , Z _{i −2}

= b

^′ _{i −2} b ^′ _{i −1} . .

2

..

Moreover, if G is bipartite, then for 0

≤ j ≤ d, a ^′ j

= 0, and

p _j

(x) is even or odd depending on whether j is even or odd.

Distance polynomials

.

1

..

Let α be the eigenvector of A corresponding to λ

0

such that α

^t

α = n and all entries are positive. Note that

α = (1,1,...,1)

^t

iff G is regular.

.

2

..

The matrix A

_i

, indexed by V G, satisfying (A

_i

)

_uv

=

{ α

u

α

v ,

if∂(u,v) = i;

0, else. is called the

i-th (weighted) distance matrix

of G.

.

3

.. A ₀

= p

₀

(A)(= I) iff G is regular.

.

4

.. A ₀

+ A

₁

+

··· + A D

= H(A) = p

₀

(A) + p

₁

(A) +

··· + p d

(A).

The spectral excess and the excess

.

1

.. p _d

(λ

0

) is called the

spectral excess

of G.

.

2

..

δ

d

:=

¹ _n

∑

i, j

(A

_d ◦ A d

)

_{i j}

is called the

excess

of G, where A

_d

:= 0 if D < d.

If G is regular, thenδ

d

is the average number of vertices to have distance d to a vertex.

The spectral excess theorem (SET)

.

Theorem 1.2

..

...

δ

d ≤ p d

(λ

0

) with equality iff G is a distance-regular graph.

[FG1997] M.A. Fiol and E. Garriga, From local adjacency polynomials to local pseudo-distance-regular graphs, J. Combin.

Theory Ser. B 71 (1997), 162–183.

Related definitions

Define

δ

i

:=1

n ∑

u,v

(A

_i ◦ A i

)

_uv ,

δ

≥i

:=δ

i

+δ

i+1

+

··· ,

p _≥i

(λ

0

) := p

i

(λ

0

) + p

i+1

(λ

0

) +

··· , p ^even

(λ

0

) := p

0

(λ

0

) + p

2

(λ

0

) +

··· ,

p ^odd _≥i

(λ

0

) := p

i

(λ

0

) + p

i+2

(λ

0

) +

···

for odd i,

A ^odd

:= A

1

+ A

3

+

···

...

Modify the proof of SET

.

Proposition 1.3

..

...

δ

≥i ≤ p ≥i

(λ

0

) with equality iff A

_≥i

= p

_≥i

(A).

.

Proposition 1.4

..

...

δ

≤i ≥ p _≤i

(λ

0

) with equality iff A

_≤i

= p

_≤i

(A).

G is bipartite

.

Proposition 1.5

..

...

If G is bipartite and

∗ ∈ {even,odd} then

δ

_≥i ^∗ ≤ p ^∗ _≥i

(λ

0

) with equality iff A

^∗ _≥i

= p

^∗ _≥i

(A).

.

Proposition 1.6

..If G is bipartite and

∗ ∈ {even,odd} then

δ

_≤i ^∗ ≥ p ^∗ _≤i

(λ

0

) with equality iff A

^∗

= p

^∗

(A).

Questions?

δ

i ≤ p i

(λ

0

) or δ

i ≥ p i

(λ

0

)?

Which graphs have the property thatδ

i

= p

_i

(λ

0

)?

The case i = 0

.

Proposition 2.1

..

...

δ

0 ≥ 1(= p 0

(λ

0

)) ( i.e. α

1 ⁴

+α

2 ⁴

+

··· +

α

n ⁴ ≥ n),

and the following are equivalent.

.

1

..

δ

0

= 1. (i.e. α

1 ⁴

+α

2 ⁴

+

··· +

α

n ⁴

= n.)

.

2

.. A ₀

= I.

.

3

.. G is regular.

.

4

..

The entries ofα are all 1.

The above inequality is fromδ

≤0 ≥ p ≤0

(λ

0

) which we mentioned earlier, but also follows from the Cauchy-Schwarz inequality

The case i = 1

A bipartite graph with bipartition V (G) = X

∪Y is biregular

if there exist distinct integers k

̸= k ^′

such that every x

∈ X has degree k,

and every y

∈ Y has degree k ^′ .

.

Proposition 3.1

..

...

δ

1 ≥ p 1

(λ

0

), and the following statements are equivalent.

(i) δ

1

= p

₁

(λ

0

) (or equivalently δ

1 k =

λ

0 ²

), (ii) A

₁

= p

₁

(A),

(iii) G is regular or G is bipartite biregular.

.

Corollary 3.2

..

...

There is no bipartite biregular graph G with exactly four distinct eigenvalues.

The idea of the proof is to modify a proof in

[DDFGG2011] C. Dalfó, E.R. van Dam, M.A. Fiol, E. Garriga and B.L. Gorissen, On almost distance-regular graphs, J. Combin.

Theory Ser. A 118 (2011), 1094–1113.

The case i = 2

.

1

..

If d = 2 thenδ

2 ≤ p 2

(λ

0

).

.

2

..

If G is regular bipartite, thenδ////

0

+δ

2 ≥ p

///////// p

0

(λ

0

)+

²

(λ

0

).

.

3

..

There is no hope to determine the order ofδ

2

and p

2

(λ

0

) uniformly.

.

Definition 4.1

..

...

Let G be a graph and i is a nonnegative integer. We say the numbers c

_i , a _i , b _i

respectively are

well-defined

in G if for any

x, y ∈ V(G) with

∂(x,y) = i, the numbers

c _i

:=

|G 1

(x)

∩ G i −1

(y)

|, a _i

:=

|G 1

(x)

∩ G i

(y)

|, b i

:=

|G 1

(x)

∩ G i+1

(y)

|,

respectively are independent of the choice of x, y.

d y

d x





 



c i b i

.

Definition 4.2

..

...

A graph G is

t-partially distance-regular

if for 2

≤ i ≤ t and the

numbers c

i , a i −1 , b i −2

are well-defined.

.

Lemma 4.3

..

...δ

⁰

= p

₀

(λ

0

) andδ

2

= p

₂

(λ

0

) iff G is 2-partially distance-regular.

.

Definition 4.4

..

...

For a connected bipartite graph G with bipartition X

∪Y, the halved graphs G ^X

and G

^Y

are the two connected components of the distance-2 graph of G.

G :

'

&

$

% G ^X

'

&

$

% G ^Y

.

Theorem 4.5 (BCN, Prop 4.2.2, p.141)

..

...

The halved graphs of a bipartite distance-regular graph are again distance-regular and, in the case where G is vertex transitive, the two halved graphs are isomorphic.

.

Lemma 4.6

..

...

If G = (X,Y ) is a connected regular bipartite graph with

δ

2

= p

₂

(λ

0

) (so 2–partially distance-regular by previous lemma), then the halved graphs G

^X

and G

^Y

have the same spectrum.

Bipartite graphs with δ d −1 = p _{d −1} ( λ 0 )

.

Lemma 5.1

..

...

Let G be bipartite andδ

d −1

= p

d −1

(λ

0

). Then

A _{d −1}

= p

_{d −1}

(A), A

_{d −3}

= p

_{d −3}

(A), A

_{d −5}

= p

_{d −5}

(A), . . . . In particular G is regular (if A

₀

= p

₀

(A)) or bipartite biregular (if

A 1

= p

₁

(A)).

.

Theorem 5.2

..

Let G be a connected bipartite graph with bipartition X

∪Y, odd d.

Then the following are equivalent.

(i)

δ

d −1

= p

_{d −1}

(λ

0

);

(ii) G is 2-partially distance-regular and both halved graphs G ^X

The following example shows that a bipartite graph satisfying Theorem 5.2(i)-(ii) and D = d needs not to be distance-regular graph.

.

Example 5.3

..

...

Consider the regular bipartite graphs G on 20 vertices obtained from the Desargues graph (the bipartite double of the Petersen graph) by the GM-switching (a way to produce cospectral

nonisomorphic graphs). One can check (by Maple) that D = d = 5, sp G =

{3 ¹ , 2 ⁴ , 1 ⁵ , (−1) ⁵ , (−2) ⁴ , (−3) ¹ }, p 0

(x) = 1, p

1

(x) = x,

p ₂

(x) = x

² − 3, p 3

(x) = (x

³ − 5x)/2, p 4

(x) = (x

⁴ − 9x ²

+ 12)/4,

p ₅

(x) = (x

⁵ − 11x ³

+ 22x)/12, A

_i

= p

_i

(A) for i

∈ {0,1,2,4}. Hence

δ

0

= p

0

(λ

0

) = 1,δ

1

= p

1

(λ

0

= 3,δ

2

= p

2

(λ

0

) = 6,

δ

4

= p

₄

(λ

0

) = 3),δ

3

= 32/5,δ

5

= 3/5 , p

₃

(λ

0

) = 6,

p ₅

(λ

0

) = 1

̸= 3/5 =

δ

5

. Then G is not distance-regular.

The following example provides a graph G satisfying Theorem 5.2(i)-(ii) with D = d

− 1.

.

Example 5.4

..Consider the Möbius-Kantor graph G, i.e., the generalized Petersen graph GP(8, 3) with vertex set

{u 0 , u ₁ , . . . , u ₇ , v ₀ , v ₁ , . . . , v ₇ } and

edge set

{u i v i , u i u i+1 , v i v i+3 |0 ≤ i ≤ 7} with arithmetic modulo 8.

One can check (by Maple) that D = 4 < 5 = d, sp G =

{3 ¹ , √

3

⁴ , 1 ³ , ( −1) ³ , ( − √

3)

⁴ , ( −3) ¹ }, p 0

(x) = 1, p

1

(x) = x,

p ₂

(x) = x

² − 3, p 3

(x) = 2(x

³ − 5x)/5, p 4

(x) = (x

⁴ − 10x ²

+ 15)/6,

p 5

(x) = (x

⁵ − 56x ³ /5 + 21x)/18, A _i

= p

_i

(A) for i

∈ {0,1,2,4}

(δ

0

= p

0

(λ

0

) = 1,δ

1

= p

1

(λ

0

) = 3, δ

2

= p

2

(λ

0

) = 6,

δ

4

= p

₄

(λ

0

) = 1),δ

3

= 5, p

₃

(λ

0

) = 24/5, p

₅

(λ

0

) = 1/5. Note that

G ²

= 2X , where X is the 16-cell graph

(http://mathworld.wolfram.com/16-Cell.html), which is distance-regular with sp X =

{6 ¹ , 0 ⁴ , ( −2) ³ }.

The parity of d makes things different

.

Theorem 5.5

..

...

Let G be a connected bipartite graph with bipartition X

∪Y and even d. Then the following are equivalent.

(i) G is distance-regular;

(ii) G is 2-partially distance-regular and both of the halved graphs G ^X

and G

^Y

are distance-regular of diameter d/2.

In the next two pages, we provide two examples of

non-distance-regular graphs that satisfy p

_d−1

(λ

0

) =δ

d−1

when d is even. The first one is bipartitle biregular and the second one is regular.

.

Example 5.6

..Consider the bipartite graphs G on 25 vertices obtained from the Petersen graph by subdividing each edge once. One can check (by Maple) that D = d = 6,

sp G =

{ √

6

¹ , 2 ⁵ , 1 ⁴ , 0 ⁵ , ( −1) ⁴ , ( −2) ⁵ , ( − √

6)

¹ }, the

Perron-Frobenius vectorα = (√

5/4,

··· ,

√

| {z 5/4}

10

,

√

5/6,

··· ,

√

| {z 5/6}

15

)

^t

,

p 0

(x) = 1, p

₁

(x) = 5

√

6x/12, p

2

(x) = 15(x

² − 12/5)/16, p 3

(x) = 5

√

6(x

³ − 4x)/12, p 4

(x) = 25(x

⁴ − 21x ² /4 + 3)/28, p ₅

(x) = 5

√

6(x

⁵ − 7x ³

+ 10x)/24,

p 6

(x) = 5(x

⁶ −65x ⁴ /7 + 22x ² −48/7)/24, e A i

= p

_i

(A) for i

∈ {1,3,5}

(δ

1

= p

1

(λ

0

) = 5/2,δ

3

= p

3

(λ

0

) = 5,δ

5

= p

₅

(λ

0

) = 5), δ

0

= 25/24, δ

2

= 85/24,δ

4

= 85/12,δ

6

= 5/6, p

₀

(λ

0

) = 1, p

₂

(λ

0

) = 27/8,

p 4

(λ

0

) = 375/56, p

₆

(λ

0

) = 10/7. Note that G

²

is the disjoint union of the Petersen graph X and the line graph Y of X. We have

.

Example 5.7

..

...

Let G be the Hoffman graph (A graph nonisomorphic but cospectral to 4-cube). Then sp G =

{4 ¹ , 2 ⁴ , 0 ⁶ , ( −2) ⁴ , ( −4) ¹ }, D = d = 4, p ₀

(x) = 1, p

₁

(x) = x, p

₂

(x) = (x

² − 4)/2,

p 3

(x) = (x

³ − 10x)/6, p 4

(x) = (x

⁴ − 16x ²

+ 24)/24, and A

₃

= p

₃

(A).

Note that G

²

is the disjoint union of K

8

and K

2,2,2,2

(= K

8 − 4K 2

), which are both distance-regular (sp K

_2,2,2,2

=

{6 ¹ , 0 ⁴ , ( −2) ³ }).