### Spectral Excess Theorem and its Applications

李光祥 (交大應數)

2014 組合數學新苗研討會 台灣師範大學

August 3, 2014

This research is conducted under the supervision of Professor Chih-wen Weng.

### Distance-regularity

*Let G be a connected graph on n vertices, with vertex set V and*
*diameter D.*

For 0*≤ i ≤ D and two vertices u,v ∈ V at distance i, set*
*c*_{i}*(u, v) : =|G*1*(v)∩ G**i−1**(u)|,*

*a**i**(u, v) : =|G*1*(v)∩ G**i**(u)|, and*
*b**i**(u, v) : =|G*1*(v)∩ G**i+1**(u)|.*

These parameters arewell-deﬁned if they are independent of the
*choice of u, v. In this case we use the symbols c**i**, a**i* *and b**i* for short.

*A connected graph G with diameter D is called*distance-regular
if the above-mentioned parameters are well-deﬁned.

*Assume that adjacency matrix A has d + 1 distinct eigenvalues*
λ0*>*λ1*> . . . >*λ*d* *with corresponding multiplicities m*_{0}*= 1, m*_{1}*, . . .,*
*m** _{d}*.

Thespectrum*of G is denoted by the multiset*
*sp G ={λ*_{0}^{m}^{0}*,*λ_{1}^{m}^{1}*, . . . ,*λ_{d}^{m}^{d}*}.*

*The parameter d is called the*spectral diameter*of G.*

*Note that D≤ d.*

**Question: Is the distance-regularity of a graph determined by its**
spectrum?

**Answer: In general, the answer is negative.**

. ...

*u*

..

*v*
..

*u*^{′}

..

*v*^{′}

The Hamming 4-cube and the Hoﬀman graph
(distance-regular) *(c*2 is not well-deﬁned)

We have known that the distance-regularity of a graph is in general not determined by its spectrum.

**Question: Under what additional conditions, the answer is positive?**

**Answer: The spectral excess theorem.**

The spectral excess theorem gives a quasi-spectral characterization for a regular graph to be distance-regular.

.Spectral excess theorem (Fiol and Garriga, 1997) ..

...

*Let G be a* regular *graph with d + 1 distinct eigenvalues. Then,*

*k**d**≤ p**d*(λ0*), and equality is attained if and only if G is distance-regular.*
*k** _{d}*=

^{1}

*∑*

_{n}*u*

*∈V*

*|G*

*d*

*(u)|: average excess (*combinatorial aspect) –the

*mean of the numbers of vertices at distance d from each vertex*

*p*

*(λ0): spectral excess (algebraic aspect) –a number which can be computed from the spectrum*

_{d}Therefore, besides the spectrum, a simplecombinatorial property suﬃces for a regular graph to be distance-regular.

. .

The Hamming 4-cube and the Hoﬀman graph
*(k*_{d}*= 1 = p** _{d}*(λ0))

*(k*

*=*

_{d}*1/2< 1 = p*

*(λ0))*

_{d}.Spectral excess theorem (Fiol and Garriga, 1997) ..

...

*Let G be a* regular *graph with d + 1 distinct eigenvalues. Then,*

*k*_{d}*≤ p**d*(λ0*), and equality is attained if and only if G is distance-regular.*

The following example shows that the spectral excess theoremcannot be directly applied to nonregular graphs.

.Example ..

...

*Let G be a path on three vertices, with spectrum* *{√*

*2, 0,−√*
2}.

*Note that D = d = 2, k*_{2}*= 2/3 and p*_{2}(λ0*) = 1/2. This shows that*
*k**d**≤ p**d*(λ0) does not hold for nonregular graphs.

Thus, a ‘weighted’ version of the spectral excess theorem is given in order to make it applicable to nonregular graphs.

### Two kinds of inner products

Consider the vector spaceR*d**[x] consisting of all real polynomials of*
*degree at most d with the inner product*

*⟨p,q⟩**G**:= tr(p(A)q(A))/n =*

### ∑

*u,v*

*(p(A)◦ q(A))**uv**/n,*

*for p, q∈ R**d**[x], where◦ is the entrywise product of matrices.*

*For any two n× n symmetric matrices M,N over R, deﬁne the inner*
product

*⟨M,N⟩ :=*1
*n*

### ∑

*i, j*

*(M◦ N)**i j**,*
where “*◦ ” is the entrywise product of matrices.*

Thus*⟨p,q⟩**G*=*⟨p(A),q(A)⟩* *for p, q∈ R**d**[x].*

### Predistance polynomials

*By the Gram–Schmidt procedure, there exist polynomials p*0*, p*1*, . . . , p**d* in
R*d**[x] satisfying*

*deg p**i**= i* and *⟨p**i**, p**j**⟩**G*=δ*i j**p**i*(λ0)
for 0*≤ i, j ≤ d, where*δ*i j**= 1 if i = j, and 0 otherwise.*

These polynomials are called thepredistance polynomials *of G.*

.Lemma (Fiol and Garriga, 1997) ..

...

*The spectral excess p** _{d}*(λ0) can be expressed in terms of the spectrum,
which is

*p** _{d}*(λ0) =

*n*π0

^{2}

( *d*
*i=0*

### ∑

1
*m** _{i}*π

*i*

^{2}

)_{−1}*,*

whereπ*i*=∏*j**̸=i**|λ**i**−*λ*j**| for 0 ≤ i ≤ d.*

### Preparation for the ‘weighted’ version

Let*α be the eigenvector of A (usually called the*Perron vector)
corresponding toλ0 such that α^{t}*α = n and all entries of α are*
positive. Note that*α = (1,1,...,1)*^{t}*iﬀ G is regular.*

*For a vertex u, let*α*u* *be the entry corresponding to u in* α.

The matrix e*A**i* *with rows and columns indexed by the vertex set V*
such that

( e*A** _{i}*)

*=*

_{uv}{α*u*α*v* if *∂(u,v) = i,*
0 otherwise
is called the *weighted distance-i matrixof G.*

*If G is regular, then eA**i* is binary and hence the *distance-i matrixA**i*.
Thus, e*A**i* *can be regarded as a ‘weighted’ version of A**i*.

### Spectral excess theorem and its ‘weighted’ version

Deﬁne eδ*i*:=*⟨eA**i**, eA**i**⟩. If G is regular, then*
eδ*d*=*⟨eA*_{d}*, eA*_{d}*⟩ = ⟨A**d**, A*_{d}*⟩ =*1

*n*

### ∑

*u**∈VG**|G**d**(u)| = k**d**.*
Hence, eδ*d* *can be viewed as a generalization of average excess k**d*.
.Spectral excess theorem (Fiol and Garriga, 1997)

..

...

*Let G be a* regular *graph with d + 1 distinct eigenvalues. Then,*

*k*_{d}*≤ p**d*(λ0), and equality is attained if and only if G is distance-regular.
.Weighted spectral excess theorem (Lee and Weng, 2012)

..

...

*Let G be a graph with d + 1 distinct eigenvalues. Then,*

eδ*d**≤ p**d*(λ0), and equality is attained if and only if G is distance-regular.

*An application: graphs with odd-girth 2d + 1*

.Odd-girth theorem (van Dam and Haemers, 2011) ..

...

A connectedregular *graph with d + 1 distinct eigenvalues and odd-girth*

*2d + 1 is distance-regular.*

In the same paper, they posed the question to determine whether the regularity assumption can be removed.

Moreover, they showed that the answer is aﬃrmative for the case
*d + 1 = 3, and claimed to have proofs for the cases d + 1∈ {4,5}.*

As an application of the ‘weighted’ spectral excess theorem, we show thatthe regularity assumption is indeed not necessary.

.Odd-girth theorem (Lee and Weng, 2012) ..

...

*A connected graph with d + 1 distinct eigenvalues and odd-girth 2d + 1 is*

distance-regular.

We then apply this line of study to the class of bipartite graphs.

Thedistance-2 graph*G*^{2} *of G is the graph whose vertex set is the*
*same as of G, and two vertices are adjacent in G*^{2} if they are of
*distance 2 in G.*

For a connected bipartite graph, the halved graphsare the two connected components of its distance-2 graph.

*For an integer h≤ d, we say that G isweighted h-punctually*
distance-regularif e*A*_{h}*= p*_{h}*(A).*

.Proposition (BCN, Proposition 4.2.2, p.141) ..

...

*Suppose that G is a connected bipartite graph. If G is distance-regular,*
then its two halved graphs are distance-regular.
.Problem (The converse statement)

..

...

*Suppose that G is a connected bipartite graph, and both halved graphs are*
distance-regular.

*Can we say that G is distance-regular?*

If not, what additional conditions do we need?

**Answer:**

Three examples will be given to show that the converse does not hold, that is, a connected bipartite graph whose halved graphs are

distance-regularmay not bedistance-regular.

We will give a quasi-spectral characterization of a connected bipartite weighted 2-punctually distance-regulargraph whose halved graphs are distance-regular.

In the casethe spectral diameter is even we show that the graph characterized above is distance-regular.

### Three counterexamples

.Example 1 (weighted 2-punctually distance-regular & odd spectral diameter) ..

...

The Möbius–Kantor graph, with spectrum*{3*^{1}*,**√*

3^{4}*, 1*^{3}*, (−1)*^{3}*, (−**√*

3)^{4}*, (−3)*^{1}*}*.
*D = 4 <5 = d,*

*A*e_{i}*= p*_{i}*(A)for i∈ {0,1,*2, 4*}, and*

*both halved graphs the complete multipartite graphs K** _{2,2,2,2}*
(with spectrum

*{6*

^{1}

*, 0*

^{4}

*, (−2)*

^{3}

*}), which are*distance-regular.

.

.Example 2 (not weighted 2-punctually distance-regular & even spectral diameter) ..

...

Consider the Hoﬀman graph with spectrum*{4*^{1}*, 2*^{4}*, 0*^{6}*, (−2)*^{4}*, (−4)*^{1}*},*
which is cospectral to the Hamming 4-cube but not distance-regular.

*D =d = 4,*

*A*e*i**= p**i**(A)for i∈ {0,1,3} (i̸= 2*), and

*its two halved graphs are the complete graph K*8 and the complete
*multipartite graph K**2,2,2,2*, which are bothdistance-regular.

. .

.Example 3 (not weighted 2-punctually distance-regular & odd spectral diameter) ..

...

Consider the graph obtained by deleting a 10-cycle from the complete
*bipartite graph K** _{5,5}*, with spectrum

*{3*^{1}*, ((**√*

*5 + 1)/2)*^{2}*, ((**√*

5− 1)/2)^{2}*, ((**−**√*

*5 + 1)/2)*^{2}*, ((**−**√*

5*− 1)/2)*^{2}*, (**−3)*^{1}*}*.
*D = 3 <5 = d,*

*A*e_{i}*= p*_{i}*(A)for i∈ {0,1} (i̸= 2*), and

*both halves graphs are the complete graphs K*_{5}, which are
distance-regular.

.

### The remaining case

We have considered three counterexamples.

Example 1 (weighted 2-punctually distance-regular & odd spectral diameter) Example 2 (not weighted 2-punctually distance-regular & even spectral diameter) Example 3 (not weighted 2-punctually distance-regular & odd spectral diameter)

Note that the remaining case is that

*G is* weighted 2-punctually distance-regular with even spectral diameter.

**Question: How about the remaining case?**

**Answer: Under these additional conditions, the converse statement is true.**

.Theorem (Lee and Weng, 2014) ..

...

*Suppose that G is a connected bipartite graph, and both halved graphs are*
*distance-regular. If G is*weighted 2-punctually distance-regular with even
spectral diameter, then G is distance-regular.

.Sketch of proof ..

...

By the weighted 2-punctually distance-regularity assumption,
*G is regular, and*

both halved graphs have the same spectrum, and thus have the same (pre)distance-polynomials.

By the above results and the even spectral diameter assumption,
δe*d**= p** _{d}*(λ0), and the result follows by (weighted) spectral excess

theorem.