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 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.
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.
Question: Is the distance-regularity of a graph determined by its spectrum?
Answer: In general, the answer is negative.
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u
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v ..
u′
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v′
The Hamming 4-cube and the Hoffman graph (distance-regular) (c2 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,
kd≤ pd(λ0), and equality is attained if and only if G is distance-regular. kd=1n∑u∈V|Gd(u)|: average excess (combinatorial aspect) –the mean of the numbers of vertices at distance d from each vertex pd(λ0): 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 = pd(λ0)) (kd=1/2< 1 = pd(λ0))
.Spectral excess theorem (Fiol and Garriga, 1997) ..
...
Let G be a regular graph with d + 1 distinct eigenvalues. Then,
kd≤ pd(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 ..
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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 p2(λ0) = 1/2. This shows that kd≤ pd(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 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].
Predistance polynomials
By the Gram–Schmidt procedure, there exist polynomials p0, p1, . . . , pd in Rd[x] satisfying
deg pi= i and ⟨pi, pj⟩G=δi jpi(λ0) 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) ..
...
The spectral excess pd(λ0) can be expressed in terms of the spectrum, which is
pd(λ0) = n π02
( d i=0
∑
1 miπi2
)−1 ,
whereπi=∏j̸=i|λi−λj| for 0 ≤ i ≤ d.
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=
{αuαv 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.
Spectral excess theorem and its ‘weighted’ version
Define eδi:=⟨eAi, eAi⟩. If G is regular, then eδd=⟨eAd, eAd⟩ = ⟨Ad, Ad⟩ =1
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≤ pd(λ0), 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,
eδd≤ pd(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) ..
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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.
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).
.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?
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) ..
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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.
. .
.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?
Answer: Under these additional conditions, the converse statement is true.
.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.
