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2013 International Conference on Combinatorics
Spectral characterization of graphs
Chih-wen Weng
Joint work with Yu-pei Huang, Guang-Siang Lee and Chia-an Liu
Department of Applied MathematicsNational Chiao Tung University
July 12, 2013
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 1 / 47
2013 International Conference on Combinatorics
Notations
Let G be a simple connected graph of order n.
The adjacency matrix A = (aij) of G is a binary square matrix of order nwith rows and columns indexed by the vertex set V G of G such that forany i, j ∈ V G, aij = 1 if i, j are adjacent in G.
d d d1 2 3
A =
0 1 01 0 10 1 0
.
Let λ1(A) ≥ λ2(A) ≥ · · · ≥ λn(A) denote the eigenvalues of A, andλi(G) := λi(A).
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 2 / 47
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Eigenvalues help us to realize the structure of a graph
.Theorem..
......For a graph G of order n, G is bipartite if and only ifλ1(G) = −λn(G).
Dongbo Bu, et al., Topological structure analysis of the protein-proteininteraction network in budding yeast, Nucleic Acids Research, 2003, Vol.31, No. 9, 2443-2450.
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 3 / 47
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Eigenvalues help us to solve problems in Combinatorics
Let χ(G) denote the chromatic number of G.
.Theorem (Wilf Theorem(1967) and Hoffman(1970))..
......
For a graph G,
(λn(G)− λ1(G))/λn(G) ≤ χ(G) ≤ λ1(G) + 1.
To estimate the integer value χ(G), only approximations of λ1(G) andλn(G) are necessary.
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 4 / 47
2013 International Conference on Combinatorics
Estimate the eigenvalues of a matrix by matrices of smallersizesIt is well-known that
λ1 = maxx∈Rnx⊤x=1
x⊤Ax, λn = minx∈Rnx⊤x=1
x⊤Ax.
The following theorem generalizes this property.
.Theorem (Cauchy interlacing theorem)..
......
For m < n, and an m× n matrix S with SS⊤ = I,
λi(A) ≥ λi(SAS⊤),
λn+1−i(A) ≤ λm+1−i(SAS⊤)
for 1 ≤ i ≤ m.
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 5 / 47
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Example
Choose S = [I 0] in block form and then SAS⊤ becomes the adjacencymatrix of an induced subgraph of G.
List the eigenvalues of paths Pn and Pn−1 of orders n and n− 1respectively:
2 cos π
n+ 1> 2 cos 2π
n+ 1> 2 cos 3π
n+ 1> · · · > 2 cos (n− 1)π
n+ 1> 2 cos nπ
n+ 1
↘ 2 cos π
n> 2 cos 2π
n> · · · > 2 cos (n− 1)π
n↗
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 6 / 47
2013 International Conference on Combinatorics
The above method does not give us an upper bound of λ1(A).
Can we find a matrix M whose largest eigenvalue λ1(M) gives an upperbound of λ1(G)?
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 7 / 47
2013 International Conference on Combinatorics
Perron-Frobenius Theorem
Let d1 ≥ d2 ≥ · · · ≥ dn denote the degree sequence of G.
.Theorem..
......
λ1(G) ≤ d1
with equality iff G is regular.
Let [d1] be a 1× 1 matrix. The above theorem says
λ1(G) ≤ λ1([d1]).
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 8 / 47
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Another upper bound of λ1(G) is.Theorem (Stanley, 1987)..
......
λ1(G) ≤−1 +
√1 + 8|EG|2
with equality if and only if G is the complete graph Kn.
Equivalently, λ1(G) is bounded above by
λ1
0 1 · · · 1 d1 − (n− 1)1 0 1 · · · 1 d2 − (n− 1)... . . . . . . . . . ...
1 0 dn − (n− 1)1 · · · 1 dn+1 − n
(n+1)×(n+1)
,
where dn+1 := 0, thinking of an isolated vertex being added.(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 9 / 47
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An improvement of Stanley Theorem is.Theorem (Yuan Hong, Jin-Long Shu and Kunfu Fang, 2001)..
......
λ1(G) ≤dn − 1 +
√(dn + 1)2 + 4(2|EG| − ndn)
2,
with equality if and only if G is regular or there exists 2 ≤ t ≤ n such thatd1 = dt−1 = n− 1 and dt = dn.
Equivalently, λ1(G) is bounded above by
λ1
0 1 · · · 1 d1 − n+ 21 0 1 · · · 1 d2 − n+ 2... . . . . . . . . . ...
1 0 dn−1 − n+ 21 · · · 1 dn − n+ 1
n×n
.
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 10 / 47
2013 International Conference on Combinatorics
Another version is.Theorem (Kinkar Ch. Das, 2011)..
......
λ1(G) ≤d2 − 1 +
√(d2 + 1)2 + 4(d1 − d2)
2,
with equality if and only if either G is regular, or d1 = n− 1 andd2 = dn.
Equivalently,
λ1(G) ≤ λ1
([0 d11 d2 − 1
]2×2
).
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 11 / 47
2013 International Conference on Combinatorics
The parameter ϕℓ
For 1 ≤ ℓ ≤ n, let
ϕℓ(G) :=λ1
0 1 · · · 1 d1 − ℓ+ 21 0 1 · · · 1 d2 − ℓ+ 2... . . . . . . . . . ...
1 0 dℓ−1 − ℓ+ 21 · · · 1 dℓ − ℓ+ 1
ℓ×ℓ
=dℓ − 1 +
√(dℓ + 1)2 + 4
∑ℓ−1i=1(di − dℓ)
2.
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 12 / 47
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.Theorem (Chia-an Liu, —, 2013)..
......
For each 1 ≤ ℓ ≤ n,λ1(G) ≤ ϕℓ(G),
with equality iff G is regular or there exists 2 ≤ t ≤ ℓ such thatd1 = dt−1 = n− 1 and dt = dn.
Moreover, we show that the function ϕℓ(G) in variable ℓ is convex.
ϕ1 r rϕ2
r r r rr ϕnr ϕn−1
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 13 / 47
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Small technical difficulty in the proof
The matrix
0 1 · · · 1 d1 − ℓ+ 21 0 1 · · · 1 d2 − ℓ+ 2... . . . . . . . . . ...
1 0 dℓ−1 − ℓ+ 21 · · · 1 dℓ − ℓ+ 1
ℓ×ℓ
needs not to be nonnegative.
Our formal proof follows the idea of Jinlong Shu and Yarong Wu 2004,which applies Perron-Frobenius Theorem to U−1AU with some carefullyselected diagonal matrix U .
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 14 / 47
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The number mi :=1di
∑j∼i dj is called the average 2-degree of i. List mi
in the decreasing ordering asM1 ≥M2 ≥ · · · ≥Mn.
t t
t
t t
t t
t t
A non-regular graph with M1 =M2 = · · · =M9 = 3
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 15 / 47
2013 International Conference on Combinatorics
Applying Perron-Frobenius Theorem tod1 0
d2. . .
0 dn
−1
A
d1 0
d2. . .
0 dn
,
we have.Theorem..
......
λ1(G) ≤M1
with equality iff M1 =Mn.
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 16 / 47
2013 International Conference on Combinatorics
An improvement of the upper bound M1,.Theorem (Ya-hong Chen and Rong-yin Pan and Xiao-dong Zhang,2011)..
......
λ1(G) ≤M2 − a+
√(M2 + a)2 + 4a(M1 −M2)
2,
with equality iff M1 =Mn, where a = max{di/dj | 1 ≤ i, j ≤ n}.
Equivalently,λ1(G) ≤ λ1
([0 M1
a M2 − a
]).
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 17 / 47
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Let b ≥ max{di/dj | 1 ≤ i, j ≤ n, i ∼ j}, and for 1 ≤ ℓ ≤ n, let
ψℓ(G) :=λ1
0 b · · · b M1 − (ℓ− 2)bb 0 b · · · b M2 − (ℓ− 2)b... . . . . . . . . . ...
b 0 Mℓ−1 − (ℓ− 2)bb · · · b Mℓ − (ℓ− 1)b
ℓ×ℓ
=Mℓ − b+
√(Mℓ + b)2 + 4b
∑ℓ−1i=1(Mi −Mℓ)
2.
.Theorem (Yu-pei Huang, —, 2013)..
......
For each 1 ≤ ℓ ≤ n,λ1(G) ≤ ψℓ(G),
with equality iff M1 =Mn.
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 18 / 47
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Problem: In the spirit of Cauchy interlacing theorem, give a uniform wayto find a matrix M with λ1(A) ≤ λ1(M) that generalizes the abovematrices.
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 19 / 47
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Sometimes, the eigenvector α > 0 (Perron vector) of A corresponding toλ1(A) also involves in the study.
For instance the Perron vector of the web graph plays a key role in rankingthe web pages by Google.
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 20 / 47
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Our second spectral characterization of graphs is related todistance-regular graphs.
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 21 / 47
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Distance-regular graphs
We recall definition of DRGs and their basic properties.
A graph G with diameter D is distance-regular if and only if for i ≤ D,
ci := |G1(x) ∩Gi−1(y)|,ai := |G1(x) ∩Gi(y)|,bi := |G1(x) ∩Gi+1(y)|
are constants subject to all vertices x, y with ∂(x, y) = i.
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 22 / 47
2013 International Conference on Combinatorics
∂(x, y) = i
dy dx����
������
��ci
ai
bi
Note that ai + bi + ci = b0 and k := b0 is the valency of G.
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 23 / 47
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Distance-Regular graphs, also called P -polynomial schemes, form animportant subclass of association schemes.
”Association schemes are the frameworks on which coding theory, designtheory and other theories developed in a unified and satisfactory way. .......There are many mathematical objects whose essence is that of associationschemes and many different names are given to the essentially the samemathematical concept: Adjacency algebra, Bose-Mesner algebra,centralizer ring, Hecke ring, Schur ring, character algebra, hypergroup,probabilistic group, etc” ——–Eiichi Bannai and Tatsuro Ito
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 24 / 47
2013 International Conference on Combinatorics
Distance matrices
The matrices that we are concerned are square matrices with rows andcolumns indexed by the vertex set V G. Let α be an eigenvector of Acorresponding to λ1(G) normalized to α⊤α = n. For each i let Ai be thematrix with entries
(Ai)xy =
{αxαy, if ∂(x, y) = i;0, else.
Ai is called i-th distance matrix of Γ. Note A0 = I and A−1 = AD+1 = 0.
If G is regular then α = (1, 1, . . . , 1)⊤, so Ai is binary and A1 = A.
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 25 / 47
2013 International Conference on Combinatorics
t t
ttt
tt t
G
A0 = I,
A1 =
0 0 0 0 1 1 1 00 0 0 0 1 1 0 10 0 0 0 1 0 1 10 0 0 0 0 1 1 11 1 1 0 0 0 0 01 1 0 1 0 0 0 01 0 1 1 0 0 0 00 1 1 1 0 0 0 0
,
A2 =
0 1 1 1 0 0 0 01 0 1 1 0 0 0 01 1 0 1 0 0 0 01 1 1 0 0 0 0 00 0 0 0 0 1 1 10 0 0 0 1 0 1 10 0 0 0 1 1 0 10 0 0 0 1 1 1 0
,
A3 =
0 0 0 0 0 0 0 10 0 0 0 0 0 1 00 0 0 0 0 1 0 00 0 0 0 1 0 0 00 0 0 1 0 0 0 00 0 1 0 0 0 0 00 1 0 0 0 0 0 01 0 0 0 0 0 0 0
.
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 26 / 47
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Three-term recurrence relation of DRGs
.Theorem..
......
Let G be a regular graph. Then the following are equivalent....1 G is distance-regular;...2 AAi = bi−1Ai−1 + aiAi + ci+1Ai+1 0 ≤ i ≤ D;...3 there exist a unique sequence of polynomials p0(x) = 1, p1(x) = x,. . ., pD(x) such that deg(pi) = i and Ai = pi(A).
The polynomials p0(x) = 1, p1(x) = x, . . ., pD(x) are called distancepolynomials of a DRG, but they can be reconstructed in a general graph.
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 27 / 47
2013 International Conference on Combinatorics
Let G a general graph G with adjacency matrix A and minimal polynomialof degree d+ 1. Since A is symmetric, A has d+ 1 distinct eigenvalues.The number d is called the spectral diameter of G. It is well-known thatd ≥ D.
Define an inner product on the space of real polynomials of degrees atmost d by
⟨f(λ), g(λ)⟩ = 1
ntrace
(f(A)g(A)⊤
).
Then there exists a unique sequence of orthogonal polynomials p0(x) = 1,p1(x), . . ., pd(x) such that
deg(pi) = i, and ⟨pi(x), pi(x)⟩ = pi(λ1).
G is t-partially distance-regular if Ai = pi(A) for 0 ≤ i ≤ t.
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 28 / 47
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The numberpd(λ1)
is called the spectral excess of G; while the number
δD :=1
ntrace(ADA
⊤D)
is called the excess of G.
When G is regularδD =
1
n
∑x∈V (G)
|GD(x)|
is the average number of vertices which have distance the diameter to avertex.
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 29 / 47
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Spectral Excess Theorem
.Theorem (M.A. Fiol, E. Garriga and J.L.A. Yebra, 1996)..
......
If G is regular thenδD ≤ pd(λ1),
with equality iff G is distance-regular.
Short proofs are given by [E.R. van Dam, 2008] and [M.A. Fiol, S. Gagoand E. Garriga, 2010].
Base on the short proofs, the regularity assumption of G is dropped in theSpectral Excess Theorem by [Guang-Siang Lee, , 2012].
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 30 / 47
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Application
The odd girth of a graph is the smallest length of an odd cycle in the thegraph.
.Corollary (E.R. van Dam and W.H. Haemers, 2011)........A regular graph with odd girth 2d+ 1 is a generalized odd graph.
The above corollary generalizes the spectral characterization of generalizedodd graphs [Tayuan Huang, 1994], [Tayuan Huang and Chao Rong Liu,1999]. Tayuan Huang is an Emeritus of NCTU.
The regularity assumption is dropped in the above corollary by[Guang-Siang Lee, , 2012].
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 31 / 47
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Applying Spectral Excess Theorem to bipartite graphs, we have
.Theorem (Guang-Siang Lee, , 2013)..
......
Assume G is bipartite with bipartition X ∪ Y and even spectral diameter d.Then the following are equivalent.(i) δD = pd(λ1);
(ii) G is distance-regular;(iii) G is 2-partially distance-regular and both of the halved graphs GX
and GY are distance-regular.
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 32 / 47
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The assumption 2-partially distance-regular is necessaryThe following example gives a regular bipartite graph G with GX = GY
being a clique and even spectral diameter, but G is not 2-partiallydistance-regular.
.Example..
......
Let G = K5,5 − C4 − C6 be a regular graph obtained by deleting a C4 anda C6 from K5,5. We have sp G = {31, 21, 12, 02, (−1)2, (−2)1, (−3)1},D = 3 < 6 = d and G2 = 2K5.
t t t t tt t t t t
C4 + C6
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 33 / 47
2013 International Conference on Combinatorics
.Example..
......
Let G be the Hoffman graph, which is a cospectral graph of 4-cubeobtained from 4-cune by applying GM-swithching of edges. Thensp G = {41, 24, 06, (−2)4, (−4)1}, D = d = 4, and
Ai = pi(A) iff i ∈ {0, 1, 3}.
Note that G2 is the disjoint union of K8 and K2,2,2,2(= K8 − 4K2), whichare both distance-regular (sp K2,2,2,2 = {61, 04, (−2)3}).
The 4-cube. The Hoffman graph.
Copy from http://en.wikipedia.org/wiki/Hoffman_graph(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 34 / 47
2013 International Conference on Combinatorics
Another drawing of 4-cube and Hoffman graph
The 4-cube
qqqq
qqqq
qqqq
The Hoffman graph
�
�
Dx
qqqq
qqqq
qqqq
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 35 / 47
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The assumption even spectral diameter is necessary
The following example gives a bipartite 2-partially distance-regular graph Gwith D = d = 5 such that GX , GY are distance-regular graphs withspectrum {61, 14, (−2)5} (the complement of petersen graph), but G isnot distance-regular.
.Example..
......
Consider the regular bipartite graphs G on 20 vertices obtained from theDesargues graph (the bipartite double of the Petersen graph) by theGM-switching. One can check (by Maple) that D = d = 5,sp G = {31, 24, 15, (−1)5, (−2)4, (−3)1}, and
Ai = pi(A) iff i ∈ {0, 1, 2, 4}.
Then G is not distance-regular.
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 36 / 47
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Desargues graph and its cospectral mate
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 37 / 47
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Near DRGs
Similar to the definition of excess, one can define
δi :=1
ntrace(AiA
⊤i ),
and want to characterize the graphs satisfying δi = pi(λ1) for some i.
Note thatAi = pi(A) ⇒ δi = pi(λ1).
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 38 / 47
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A bipartite graph with bipartitin V (G) = X ∪ Y is biregular if there existdistinct integers k ̸= k′ such that every x ∈ X has degree k, and everyy ∈ Y has degree k′.
.Proposition..
......
Let G be a connected graph. Then δ1 ≥ p1(λ1), and the followingstatements are equivalent.(i) δ1 = p1(λ1),
(ii) A1 = p1(A),(iii) G is regular or G is bipartite biregular.
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 39 / 47
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.Theorem (Guang-Siang Lee, , 2013)..
......
Let G be a connected bipartite graph with bipartition X ∪ Y and assumethat the spectral diameter d is odd. Then the following are equivalent.(i) δi = pi(λ1) for even i;(ii) δd−1 = pd−1(λ1);(iii) G is 2-partially distance-regular and both of the halved graphs GX
and GY are distance-regular ⌊d/2⌋.
We shall provide two graphs that satisfy the above equivalent conditions,but are not distance-regular graphs.
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 40 / 47
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We saw the first one before.
.Example (W.H. Haemers and E. Spence, 1995)..
......
Consider the regular bipartite graphs G on 20 vertices obtained from theDesargues graph (the bipartite double of the Petersen graph) by theGM-switching. One can check (by Maple) that D = d = 5,sp G = {31, 24, 15, (−1)5, (−2)4, (−3)1}, and
Ai = pi(A) iff i ∈ {0, 1, 2, 4}.
Then G is not distance-regular.
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 41 / 47
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Desargues graph and its cospectral mate
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 42 / 47
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.Example (D. Marušič and T. Pisanski, 2000)..
......
Consider the Möbius-Kantor graph G. One can check (by Maple) thatD = 4 < 5 = d, and
Ai = pi(A) iff i ∈ {0, 1, 2, 4}.
Möbius-Kantor graph
Copy from https://en.wikipedia.org/wiki
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References I...1 Ya-hong Chen and Rong-yin Pan and Xiao-dong Zhang, Two sharp
upper bounds for the signless Laplacian spectral radius of graphs,Discrete Mathematics, Algorithms and Applications, Vol. 3, No. 2(2011), 185-191.
...2 Kinkar Ch. Das, Proof of conjecture involving the second largestsignless Laplacian eigenvalue and the index of graphs, Linear Algebraand its Applications, 435 (2011), 2420-2424.
...3 Yuan Hong, Jin-Long Shu and Kunfu Fang, A sharp upper bound ofthe spectral radius of graphs, Journal of Combinatorial Theory, SeriesB 81 (2001), 177-183.
...4 Jinlong Shu and Yarong Wu, Sharp upper bounds on the spectralradius of graphs, Linear Algebra and its Applications, 377 (2004),241-248.
...5 Richard. P. Stanley, A bound on the spectral radius of graphs with eedges, Linear Algebra and its Applications, 87 (1987), 267-269.
(Dep. of A. Math., NCTU) Spectral Characterization of Graphs July 12, 2013 44 / 47
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References II...1 E.R. van Dam, The spectral excess theorem for distance-regular
graphs: a global (over)view, Electron. J. Combin. 15 (1) (2008),#R129.
...2 E.R. van Dam and W.H. Haemers, An odd characterization of thegeneralized odd graphs, J. Combin. Theory Ser. B 101 (2011),486-489.
...3 M.A. Fiol, E. Garriga and J.L.A. Yebra, On a class of polynomials andits relation with the spectra and diameters of graphs, J. Combin.Theory Ser. B 67 (1996), 48-61.
...4 M.A. Fiol, S. Gago and E. Garriga, A simple proof of the spectralexcess theorem for distance-regular graphs, Linear Algebra and itsApplications, 432(2010), 2418-2422.
...5 W.H. Haemers and E. Spence, Graphs cospectral with distance-regulargraphs, Linear Multili. Alg. 39 (1995), 91-107.
...6 D. Marušič and T. Pisanski, The remarkable generalized Petersengraph G(8, 3), Math. Slovaca 50 (2000), 117-121.
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References III
...1 Yu-pei Huang and Chih-wen Weng, Spectral Radius and Average2-Degree Sequence of a Graph, preprint.
...2 Guang-Siang Lee and Chih-wen Weng, A spectral excess theorem fornonregular graphs, Journal of Combinatorial Theory, Series A,119(2012), 1427-1431.
...3 Guang-Siang Lee and Chih-wen Weng, A characterization of bipartitedistance-regular graphs, preprint.
...4 Chia-an Liu and Chih-wen Weng, Spectral radius and degree sequenceof a graph, Linear Algebra and its Applications, (2013), http://dx.doi.org/10.1016/j.laa.2012.12.016
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Thanks for your attention.
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