# Spectral Excess Theorem and its Applications

## Full text

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### Spectral Excess Theorem and its Applications

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.

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### 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 ci(u, v) : =|G1(v)∩ Gi−1(u)|,

ai(u, v) : =|G1(v)∩ Gi(u)|, and bi(u, v) : =|G1(v)∩ Gi+1(u)|.

These parameters arewell-deﬁned if they are independent of the choice of u, v. In this case we use the symbols ci, ai and bi for short.

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

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Assume that adjacency matrix A has d + 1 distinct eigenvalues λ0>λ1> . . . >λd with corresponding multiplicities m0= 1, m1, . . ., md.

Thespectrumof G is denoted by the multiset sp G ={λ0m0,λ1m1, . . . ,λdmd}.

The parameter d is called thespectral diameterof G.

Note that D≤ d.

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

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The Hamming 4-cube and the Hoﬀman graph (distance-regular) (c2 is not well-deﬁned)

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

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

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

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

kd≤ pd0), and equality is attained if and only if G is distance-regular.  kd=1nu∈V|Gd(u)|: average excess (combinatorial aspect) –the mean of the numbers of vertices at distance d from each vertex pd0): spectral excess (algebraic aspect) –a number which can be computed from the spectrum

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

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. .

The Hamming 4-cube and the Hoﬀman graph (kd= 1 = pd0)) (kd=1/2< 1 = pd0))

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

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

kd≤ pd0), 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 ..

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

2, 0,−√ 2}.

Note that D = d = 2, k2= 2/3 and p20) = 1/2. This shows that kd≤ pd0) 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.

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### Two kinds of inner products

Consider the vector spaceRd[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∈ Rd[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∈ Rd[x].

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### Predistance polynomials

By the Gram–Schmidt procedure, there exist polynomials p0, p1, . . . , pd in Rd[x] satisfying

deg pi= i and ⟨pi, pjGi jpi0) for 0≤ i, j ≤ d, whereδi j= 1 if i = j, and 0 otherwise.

These polynomials are called thepredistance polynomials of G.

.Lemma (Fiol and Garriga, 1997) ..

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The spectral excess pd0) can be expressed in terms of the spectrum, which is

pd0) = n π02

( d i=0

### ∑

1 miπi2

)−1 ,

whereπi=∏j̸=iiλj| for 0 ≤ i ≤ d. 

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### Preparation for the ‘weighted’ version

Letα be the eigenvector of A (usually called thePerron vector) corresponding toλ0 such that αtα = n and all entries of α are positive. Note thatα = (1,1,...,1)t iﬀ G is regular.

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

The matrix eAi with rows and columns indexed by the vertex set V such that

( eAi)uv=

uαv if ∂(u,v) = i, 0 otherwise is called the weighted distance-i matrixof G.

If G is regular, then eAi is binary and hence the distance-i matrixAi. Thus, eAi can be regarded as a ‘weighted’ version of Ai.

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n

### ∑

u∈VG|Gd(u)| = kd. Hence, eδd can be viewed as a generalization of average excess kd. .Spectral excess theorem (Fiol and Garriga, 1997)

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

kd≤ pd0), and equality is attained if and only if G is distance-regular.  .Weighted spectral excess theorem (Lee and Weng, 2012)

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

d≤ pd0), and equality is attained if and only if G is distance-regular. 

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

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

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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) ..

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

distance-regular. 

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

Thedistance-2 graphG2 of G is the graph whose vertex set is the same as of G, and two vertices are adjacent in G2 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 eAh= ph(A).

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

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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)

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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?

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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.

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### Three counterexamples

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

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The Möbius–Kantor graph, with spectrum{31,

34, 13, (−1)3, (−

3)4, (−3)1}. D = 4 <5 = d,

Aei= pi(A)for i∈ {0,1,2, 4}, and

both halved graphs the complete multipartite graphs K2,2,2,2 (with spectrum{61, 04, (−2)3}), which aredistance-regular.

.

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

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Consider the Hoﬀman graph with spectrum{41, 24, 06, (−2)4, (−4)1}, which is cospectral to the Hamming 4-cube but not distance-regular.

D =d = 4,

Aei= pi(A)for i∈ {0,1,3} (i̸= 2), and

its two halved graphs are the complete graph K8 and the complete multipartite graph K2,2,2,2, which are bothdistance-regular.

. .

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

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Consider the graph obtained by deleting a 10-cycle from the complete bipartite graph K5,5, with spectrum

{31, ((

5 + 1)/2)2, ((

5− 1)/2)2, ((

5 + 1)/2)2, ((

5− 1)/2)2, (−3)1}. D = 3 <5 = d,

Aei= pi(A)for i∈ {0,1} (i̸= 2), and

both halves graphs are the complete graphs K5, which are distance-regular.

.

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### 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?

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.Theorem (Lee and Weng, 2014) ..

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

.Sketch of proof ..

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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, δed= pd(λ0), and the result follows by (weighted) spectral excess

theorem. 

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