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Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries Shanghai Jiaotong University On the Generalized Spectral Characterization of Graphs (I): The Basics Wei Wang Xian Jiaotong
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Wei Wang Xi’an Jiaotong University
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
1
Introduction
2
Notations and Terminologies
3
A Simple Characterization of DGS Graphs
4
The Method
5
Examples
6
Summaries
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Graph Isomorphism Problem (GIP) Given two graphs G and H, determine whether they are isomorphic nor not.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
There is a quasipolynomial time algorithm for all graphs, i.e., one with running time exp((log n)O(1)).a
W.X. Du (2016) The running time can be improved to nC log n for some constant C.a
aW.X, Du, On the Automorphism Group of a Graph, arXiv:1607.00547v1.
It remains an unsolved question whether GIP is in P or in NPC.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
The spectrum of a graph encodes a lot of information about the given graph, e.g., From the adjacency spectrum, one can deduce
(i) the number of vertices, the number of edges; (ii) the number of triangles ; (iii) the number of closed walks of any fixed length; (iv) bipartiteness; . . .
From the Laplacian spectrum, one can deduce:
(i) the number of spanning trees; (ii) the number of connected components; . . .
Can a graph be determined by the spectrum?
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
q q q q q q q q q q ❍❍❍❍❍❍ ✟✟✟✟✟✟
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 spectrum: -2, 0, 0, 0, 2
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
“Which graphs are determined by spectrum(DS for short)?” This is a fundamental question in Spectral Graph Theory that dates back to more than 60 years. In 1956, G¨ unthard and Primas raised the question in a paper that relates the theory of graph spectra to H¨ uckel’s theory from chemistry. Applications:
Graph Isomorphism Problem; The shape and sound of a drum; Energy of hydrocarbon molecules; . . . . . . . . .
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
To what extent a graph is DS? Are almost all graph DS? Are Almost all graphs non-DS? or Neither is true? Conjecture (Haemers) Almost all graphs are DS (w.r.t. adjacency spectrum, the Laplacian spectrum etc.) Remark: Formally speaking, the fraction of the DS graphs among all graphs tends to 1 as the order of the graphs tends to infinity.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Schwenk (1973) Almost every tree has a cospectral mate. a
aA.J. Schwenk, Almost all trees are cospectral, New Directions in The
Theory of Graphs, F. Harary (Ed.), Academic Press, New York (1973), pp. 275-307
Remark: Schwenk’s result holds for the adjacency spectrum, Laplcian spectrum, and etc.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Cospectrally rooted graph Let G and H be two graphs with root u and v respectively. Then G and H are said to be cospectrally rooted, if Spec(G) = Spec(H) and Spec(G − u) = Spec(H − v). Theorem Let G and H be two cospectrally rooted graphs with root u and v
are cospectral, where the new graphs are obtained from the old
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
16 squares as vertices; adjacent if in same row, same column, or same color. The pair of strongly regular graphs constructed in this way are cospectral. There are lots of Latin squares.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Godsil and McKay (1982), GM-switching Let the vertex set V of G can be partitioned into V and V2. Suppose G[V1] is regular, and every vertex in V2 is adjacent to non, all, or exactly half number of vertices in V1. Then the new graph obtained by GM-switching and the old one are cospectral.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
1 The complete graph Kn. 2 The regular complete bipartite graph Kn,n. 3 The cycle Cn. 4 The path Pn. 5 The tree Zn. 6 · · · · · · · · ·
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Godsil and McKay (1982); Haemers and Spence (2004); Brouwer and Haemers (2009).
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
1 A new method; 2 A simple arithmetic criterion; 3 A further development; 4 Some explicit constructions.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Two graphs G, H are cospectral w.r.t. the generalized spectrum if Spec(G) = Spec(H) and Spec(¯ G) = Spec(¯ H). E.g.
q q q q q q q ❜❜ ❜ ❜❜ ❜ ✧✧ ✧ ✧✧ ✧ q q q q q q q
❅ ❅ ❅
PG1(λ) = PG2(λ) = λ7 − 6λ5 + 9λ3 − 4λ P¯
G1(λ) = P¯ G2(λ) = λ7 − 15λ5 − 2λ4 + 12λ3 + 24λ2
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
A graph G is said to be determined by the generalized spectrum (DGS for short), if any graph that is cospectral with G w.r.t. the generalized spectrum is isomorphic to G. In notation, G is DGS if Spec(G) = Spec(H) and Spec(¯ G) = Spec(¯ H) implies H is isomorphic to G for any H.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
The walk-matrix of graph G: W (G) = [e, A(G)e, . . . , A(G)n−1e] where e = (1, 1, . . . , 1)T is the all-one vector. The (i, j)-th entry of W is the number of walks of legth j − 1 starting from the i-th vertex.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
A graph G is called a controllable graph if the corresponding walk-matrix W (G) is non-singular. The set of all controllable graphs of order n is denoted by Gn. It was conjectured (by C.D. Godsil) that almost all graphs are controllable. O
′Rourke and Touri showed recently that this conjecture is
true.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Theorem 1. [Wang and Xu, 2006] Let G ∈ Gn. Then there exists a graph H such that Spec(G) = Spec(H) and Spec(¯ G) = Spec(¯ H) if and only if there exists a unique rational orthogonal matrix Q such that QTA(G)Q = A(H), and Qe = e, (1) where e is the all-ones vector.a
determined by their generalized spectra, European J. Combin., 27 (2006) 826-840.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Lemma Let G be a graph with adjacency matrix A. Let Ak = (a(k)
ij ). Then
a(k)
ij
equals the number of walks of length k in G starting from i and ending at j. Lemma Let Nk(G) be the number of walks of length k in G. Then Nk(G) = eTAe.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Lemma Let HG(t) = ∞
k=0 Nk(G)tk be the generating function of Nk(G).
Then HG(t) = 1 t [(−1)n Φ¯
G(− 1+t t )
ΦG( 1
t )
− 1]. Corollary Spec(G) = Spec(H) and Spec(¯ G) = Spec(¯ H) implies W T
G WG = W T H WH.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Proof. “ ⇒ ” Eq. (1)⇒ Q(J − AG − I)Q = J − AH − I ⇒ QTA¯
GQ = A ¯ H.
“ ⇐ ” AGWG = AG[e, AGe, · · · , An
Ge]
= [AGe, A2
G, · · · , An Ge]
= [AGe, A2
Ge, · · · , An−1 G
e, −a1An−1
G
e − a2An−2
G
e − · · · − ane] = WG · · · −an 1 · · · −an−1 1 · · · −an−2 . . . . . . · · · ... . . . . . . . . . . . . . . . ... −a2 · · · 1 .
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Proof. Similarly, we have AHWH = WH · · · −an 1 · · · −an−1 1 · · · −an−2 . . . . . . · · · ... . . . . . . . . . . . . . . . ... −a2 · · · 1 . It follows that W −1
G AGWG = W −1 H AHWH. Let Q = WGW −1 H .
Then Q is a rational orthogonal matrix, since W T
G WG = W T H WH,
which implies QTQ = I.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Definition Γ(G) = {Q ∈ On(Q)|QTA(G)Q is a (0, 1) − matrix and Qe = e}, whereOn(Q) is the set of all rational orthogonal matrices. Theorem 2. [Wang and Xu, 2006] Let G ∈ Gn. Then G is DGS if and only if Γ(G) contains only permutation matrices. Question: How to find out all Q ∈ Γ(G) explicitly?
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Definition Let Q be a rational orthogonal matrix with Qe = e, the level of Q is the smallest positive integer ℓ such that ℓQ is an integral matrix. If ℓ = 1, then Q is a permutation matrix. Example: 1 2
1 1 1 1 −1 1 1 1 1 −1 1 1 1 1 −1
3
2 1 1 1 1 1 −1 2 −1 1 1 1 −1 −1 2 1 1 1 1 1 1 2 −1 −1 1 1 1 −1 2 −1 1 1 1 −1 −1 2
.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Definition An integral matrix M is called unimodular, if det(M) = ±1. Theorem Let M be an n by n integral matrix with full rank. Then there exist two unimodular matrices U and V such that M = U d1 d2 ... dn V , where di|di+1 for i = 1, 2, · · · , n − 1. di is called the i-th elementary divisor of M.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
An Important Fact Let Q ∈ Γ(G) with level ℓ. Then ℓ is always a divisor of the n-th elementary divisor of W (G), i.e. ℓ|dn, and hence ℓ| det(W ). Proof. This is because: QTA(H)Q = A(H), Qe = e
⇒ QTW (G) = W (H). It follows from W (G) = Udiag(d1, d2, · · · , dn)V that QT = W (H)V −1diag(d−1
1 , d−1 2 , · · · , d−1 n ))U−1 and dnQ is an
integral matrix.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Lemma (Wang and Xu, 2006) Let W be the walk-matrix of a graph G ∈ Gn. Let Q ∈ Γ(G) with level ℓ. If prime p|ℓ, then the following system of congruence equations has a non-trivial solution: W Tx ≡ 0, xTx ≡ 0 (mod p). (2) Proof. QTAGQ = AH and Qe = e imply QTWG = WH, i.e., W T
G Q = W T H . Let x be a column of Q such that x ≡ 0 (mod p)
(it always exists by the definition of ℓ). With such an x, we have W Tx ≡ 0 (mod p) and xTx ≡ 0 (mod p).
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
All the possible prime divisors of ℓ is finite; they are the divisors of dn, and hence are divisors of det(W (G)) Some of the prime divisors of det(W (G)) may not be divisors
If all the prime factors of det(W (G)) are not divisors of ℓ, then we must have ℓ = 1, and hence G is DGS.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
1 Compute det(W ) = pα1
1 pα2 2 · · · pαs s .
2 For every prime pi, if Eq. (2) has no non-trivial solution, then
p ∤ ℓ.
3 If pi ∤ ℓ, ∀i = 1, 2, · · · , s, then ℓ = 1 and G is DGS.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Let p > 2 be a prime and p|dn. If Eq. (2) has no solution, then p is not a divisor of ℓ. Assume rankp(W (G)) = n − 1. Then the solution to the system of linear equations can be written as x = kξ over finite field Fp. If ξTξ = 0, then p is not a divisor of ℓ. Using this way, the odd prime divisors of dn can be excluded in most cases.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
It can be computed dn(G1) = 2 × 17 × 67 × 8054231. For p = 17, ξTξ = 12. For p = 67, ξTξ = 25. For p = 8054231, ξTξ = 1492735. Thus, all primes (except p = 2) can be excluded.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
It can be computed dn(G2) = 2 × 32 × 5 × 197 × 263 × 5821. For p = 3, ξTξ = 1. For p = 5, ξTξ = 0. For p = 197, ξTξ = 139. For p = 263, ξTξ = 101. For p = 5821, ξTξ = 4298. Thus, all primes (except p = 2, 5) can be excluded.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
When p = 2, however, the system of linear equations has always non-trivial solutions. Thus, p=2 cannot be excluded using above method. To exclude p = 2, we more intensive exclusion conditions are needed. To conclude, it can be shown that the first graph G1 is DGS. But G2 cannot be shown to be DGS, since p = 5 cannot be excluded by using the existing methods.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Definition Q is partly decomposable, if there exist two permutation matrix P1 and P2 such that P1QP2 = Q1 O O Q2
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Theorem (Wang and Xu, 2006) Let Q be a fully indecomposable rational orthogonal matrix with level two. Then Q can be changed into the following for by permuting rows and columns: (i) 1
2
−1 1 1 1 1 −1 1 1 1 1 −1 1 1 1 1 −1 . (ii) 1
2
X O O · · · O Y Y X O · · · O O O Y X · · · O O . . . . . . . . . ... . . . . . . O O O · · · X O O O O · · · Y X , where X = 1 1 1 1
−1 −1 1
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Theorem (Wang and Xu, 2006) (iii) 1
2
1 1 1 −1 1 1 −1 1 1 1 1 −1 −1 1 1 1 1 −1 1 1 −1 1 1 1 −1 1 1 1
1 2
1 1 1 −1 1 1 −1 1 1 1 1 −1 −1 1 1 1 1 −1 1 1 −1 1 1 1 1 −1 1 1 1 1 1 −1 .
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Define Hn = {G ∈ Gn|Eq. (2) has only trivial solution for every odd p and the SNF is diag(1, 1, · · · , 1
, 2, 2, · · · , 2b
)}. Suppose G ∈ Hn. Then ℓ = 1 or ℓ = 2. If we can eliminate the possibility that ℓ = 2, then ℓ = 1 and G is DGS. But how? ℓ = 2 implies that every column of a fully indecomposable block of Q has four non-zero entry “1”, which satisfies the equation W Tx ≡ 0 (mod 2). Finding the set S of all the solutions to the above equation with exactly four “1” (say, by enumerating all n
2
cases). Try to find all Q’s with column vectors from S (modulo 2) and possibly standard unit vectors.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
Theorem Suppose G ∈ Hn. Let S be the set of solutions to the following equations: W Tx ≡ 0 (mod 2), x has exactly four“1”. If S = ∅, then G is DGS. Remark: The larger n is, the easier the above condition holds.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
n = 36, G ∈ Hn and S = ∅. Thus G is DGS.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
A = 1 1 1 1 1 1 1 1 1 1 1 1 , W = 1 1 3 6 16 35 1 3 6 16 35 86 1 2 6 13 32 73 1 3 7 16 38 87 1 2 4 9 20 47 1 1 2 4 9 20 . det(W ) = −23.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
A basis of the solutions to W Tx ≡ 0 (mod2): 1 1 1 1 1 1 ⇒ 1 1 1 1 1 1 1 1 1 1 1 1
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
1 1 1 1 ⇒ Q1 = 2 −1 1 1 1 1 −1 1 1 2 1 1 −1 1 1 1 1 −1
⇒ QT
1 AQ1 =
1 − 1
2 1 2
1
1 2 1 2
1
1 2 1 2
1
1 2 3 2
− 1
2 1 2 1 2 1 2 1 2 1 2 1 2 3 2
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
1 1 1 1 ⇒ Q2 = −1 1 1 1 1 −1 1 1 2 1 1 −1 1 2 1 1 1 −1 /2 ⇒ QT
2 AQ2 =
1 2 1 2
1 1
1 2
−1
1 2
1
1 2 1 2 1 2 1 2
1 1 1 1 1 1
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
1 1 1 1 ⇒ Q3 = 2 −1 1 1 1 1 −1 1 1 2 1 1 −1 1 1 1 1 −1 /2 ⇒ QT
3 AQ3 =
1 − 1
2 1 2
1
1 2 1 2
1
1 2 1 2
1
1 2 3 2
− 1
2 1 2 1 2 1 2 1 2 1 2 1 2 3 2
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
1 1 1 1 1 1 1 1 1 1 1 1 ⇒ Q4 = 1 −1 1 1 1 −1 1 1 1 1 1 −1 −1 1 1 1 1 1 −1 1 −1 1 1 1 /2 ⇒ QT
4 AQ4 =
−1
1 2 1 2 1 2 1 2
1
1 2 1 2 1 2 1 2 1 2 1 2
1 1
1 2 1 2
−1 1
1 2 1 2
1
1 2 1 2
1
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
1 L. Babai, Graph isomorphism in quasipolynomial time,
arXiv:1512.03547v2.
2 E. R. van Dam, W. H. Haemers, Which graphs are determined
by their spectrum? Linear Algebra Appl., 373 (2003) 241-272.
3 E. R. van Dam, W. H. Haemers, Developments on spectral
characterizations of graphs, Discrete Mathematics, 309 (2009) 576-586.
4 W.X, Du, On the Automorphism Group of a Graph,
arXiv:1607.00547v1.
5 W. Wang, C. X. Xu, A sufficient condition for a family of
graphs being determined by their generalized spectra, European J. Combin., 27 (2006) 826-840.
6 W. Wang, C.X. Xu, An excluding algorithm for testing
whether a family of graphs are determined by their generalized spectra, Linear Algebra and its Appl., 418 (2006) 62-74.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries
5 W. Wang, On the Spectral Characterization of Graphs, Phd
Thesis, Xi’an Jiaotong University, 2006.
6 W. Wang, Generalized spectral characterization of graphs
revisited, The Electronic Journal Combinatorics, 20 (4) (2013), #P4.
7 W. Wang, A simple arithmetic criterion for graphs being
determined by their generalized spectra, JCTB, to appear.
8 W. Wang, T. Yu, Square-free Discriminants of Matrices and
the Generalized Spectral Characterizations of Graphs, arXiv:1608.01144
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries