Shanghai Jiaotong University On the Generalized Spectral - PowerPoint PPT Presentation

The Theorem The Case p > 2 The Case p = 2 Examples Future Work Shanghai Jiaotong University On the Generalized Spectral Characterization of Graphs (II): A Simple Arithmetic Criterion Wei Wang Xi’an Jiaotong University Dec. 2016

The Theorem The Case p > 2 The Case p = 2 Examples Future Work Outline The Theorem 1 The Case p > 2 2 The Case p = 2 3 Examples 4 Future Work 5

The Theorem The Case p > 2 The Case p = 2 Examples Future Work Some Recent Developments Question: Does there exist a simple way to determine whether a graph is DGS ?

The Theorem The Case p > 2 The Case p = 2 Examples Future Work A Conjecture Conjecture (Wang, 2006) Define a family a graphs: F n = { G ∈ G n | det( W ) 2 [ n / 2] is an odd square − free integer } Then every graph in F n is DGS.

The Theorem The Case p > 2 The Case p = 2 Examples Future Work Remarks det( W ) is always an integer for every graph. 2 [ n / 2] G ∈ F n iff the walk-matrix W has the following SNF:   1 ...       1   ,   2     ...     2 b where b is an odd square-free integer and the number of 2 that appearers in the diagonal is [ n / 2].

The Theorem The Case p > 2 The Case p = 2 Examples Future Work Recent Resolution of The Conjecture Theorem (Wang, 2017) Conjecture 1 is true. W. Wang, A simple arithmetic criterion for graphs being determined by their generalized spectra, JCTB, 122 (2017) 438-451

The Theorem The Case p > 2 The Case p = 2 Examples Future Work The Proof: Basic Ideas For every graph G ∈ F n , let Q ∈ Γ( G ) with level ℓ , if we can show that every prime divisor of d n is not a divisor of ℓ , Then we must have ℓ = 1, and hence Q is a permutation matrix. This shows that G is DGS. This will be done in two steps: for p > 2 and p = 2.

The Theorem The Case p > 2 The Case p = 2 Examples Future Work Excluding primes p > 2 Theorem A (Wang, 2013) Let G ∈ G n . Let Q ∈ Γ( G ) with level ℓ . Suppose that p | det( W ) and p 2 � | det( W ), where p is an odd prime. Then p is not a divisor of ℓ . W. Wang, Generalized spectral characterization of graphs revisited, The Electronic Journal Combinatorics, 20 (4) (2013),# P 4.

The Theorem The Case p > 2 The Case p = 2 Examples Future Work The Proof: A sketch

The Theorem The Case p > 2 The Case p = 2 Examples Future Work Some Lemmas Lemma Let M = U diag ( d 1 , d 2 , · · · , d n ) V = USV , where S is the SNF of M and U and V are unimodular matrices, respectively. Then the system of congruence equation Mx ≡ 0 ( mod p 2 ) has a solution with x �≡ 0 ( mod p ) iff p 2 | d n . By the above lemma, it suffices for us to show that, under the conditions of the theorem, if Q ∈ Γ( Q ) with level ℓ and p | ℓ , then he congruence W T x ≡ 0 ( mod p 2 ) always has a solution with x �≡ 0 ( mod p ). Then p 2 | det( W ) and we reach a contradiction.

The Theorem The Case p > 2 The Case p = 2 Examples Future Work Some Lemmas Lemma Let Q ∈ Γ( Q ) with level ℓ and p | ℓ , then z T ( Az − λ 0 z ) ≡ 0 ( mod p 2 ) for some λ 0 and z, which satisfy W T z ≡ 0 ( mod p ) , z T z ≡ 0 ( mod p ) , and Az ≡ λ 0 z ( mod p ) . We will show that z T ( Az − λ 0 z ) ≡ 0 ( mod p 2 ) implies that p 2 | det( W ) (and in fact, the converse is also true).

The Theorem The Case p > 2 The Case p = 2 Examples Future Work The proof: A sketch The fact Q T AQ = B ( B is the adjacency matrix of some graph H ) implies that there exists a column u of ℓ Q with u �≡ 0 ( mod p ) such that u T Au = 0; u T u = ℓ 2 ; e T u = ℓ. Such a u clearly satisfies u T Au ≡ 0 ( mod p 2 ), u T u ≡ 0 ( mod p 2 ), W T u ≡ 0 ( mod p ), and e T u ≡ 0 ( mod p ). Clearly u T ( Au − λ 0 u ) ≡ 0 ( mod p 2 ) holds.

The Theorem The Case p > 2 The Case p = 2 Examples Future Work The proof: A sketch - contd. The assumption p || det( W ) implies rank p ( W ) = n − 1. It follows that rank p ( ℓ Q ) = 1, since any column of ℓ Q satisfies W T x ≡ 0 ( mod p ). Then AQ = QB and rank p ( ℓ Q ) = 1 give Au ≡ λ 0 u ( mod p ) for some integer λ 0 . Let z ≡ u + p β ( mod p ). Then z satisfies W T z ≡ 0 ( mod p ), Az ≡ λ 0 z ( mod p ), e T z ≡ 0 ( mod p ) and z T z ≡ 0 ( mod p ). It is easy to verify that the equation z T ( Az − λ 0 z ) ≡ 0 ( mod p 2 ) is invariant when replacing z with its congruence class. z T ( Az − λ 0 z ) ( u + p β ) T [( A ( u + p β ) − λ 0 ( u + p β )] ≡ u T ( Au − λ 0 u ) + 2 p β T ( Au − λ 0 u ) ≡ u T ( Au − λ 0 u ) ≡ 0 ( mod p 2 ) . ≡

The Theorem The Case p > 2 The Case p = 2 Examples Future Work Some Lemmas Lemma If rank p ( W ) = n − 1 , then rank p ( A − λ 0 I ) = n − 1 or n − 2 . Lemma If rank p ( A − λ 0 I ) = n − 2 , then rank p ([ A − λ 0 I , z ]) = n − 1 . Lemma There exists a vector y with e T y �≡ 0 ( mod p ) such that ] T ( mod p ) . W T y ≡ e T y [1 , λ 0 , · · · , λ n − 1 0

The Theorem The Case p > 2 The Case p = 2 Examples Future Work Proof of Theorem A: a sketch Q ∈ Γ( G ) implies there exists a graph H with Spec ( H ) = Spec ( G ) and Spec (¯ H ) = Spec (¯ G ). It follows that z T ( Az − λ 0 z ) ≡ 0 ( mod p 2 ), i.e., z T Az − λ 0 z ≡ 0 ( mod p ). p Case 1. If rank p ( A − λ 0 I ) = n − 1, then z T [ A − λ 0 I , Az − λ 0 z ] = 0 and rank p ( A − λ 0 I ) = n − 1 imply p that Az − λ 0 z is the linear combinations of the columns of p A − λ 0 , i.e., ( A − λ 0 I ) x ≡ Az − λ 0 z ( mod p ) for some x . p Case 2. If rank p ( A − λ 0 I ) = n − 2, then z T [ A − λ 0 I , z , Az − λ 0 z ] = 0 and rank p ([ A − λ 0 I , z ]) = n − 1 p imply that ( A − λ 0 I ) x + kz ≡ Az − λ 0 z ( mod p ) for some p vector x and integer k . Combing Cases 1 and 2, there always exists an integer m and a vector x such that ( A − λ 0 I ) x + mz ≡ Az − λ 0 z ( mod p ), i.e., p ( A − λ 0 I )( z − px ) ≡ mpz ( mod p 2 ) .

The Theorem The Case p > 2 The Case p = 2 Examples Future Work Proof of Theorem A: a sketch -contd. Left-Multiplying both sides by A i − 1 + λ 0 A i − 2 + · · · + λ i − 2 A + λ i − 1 I , we get 0 0 ( A i − λ i 0 I )( z − px ) ≡ mp ( A i − 1 + · · · + λ i − 1 I ) z ( mod p 2 ) , 0 which implies 0 e T )( z − px ) ≡ mpe T ( A i − 1 + · · · + λ i − 1 ( e T A i − λ i I ) z ≡ 0 ( mod p 2 ) , 0 for i = 0 , 1 , · · · , n − 1. That is, ] T ( mod p 2 ). W T ( z − px ) ≡ e T ( z − px )[1 , λ 0 , · · · , λ n − 1 0 Moreover, there exists a vector y such that ] T ( mod p ). W T y ≡ e T y [1 , λ 0 , · · · , λ n − 1 0 Let e T ( z − px ) ≡ pl ( mod p ) for some l . It follows that W T ( z − px − p ly e T y ) ≡ 0 ( mod p 2 ) .

The Theorem The Case p > 2 The Case p = 2 Examples Future Work Excluding prime p=2 Theorem B. [Wang, 2017] Let G ∈ F n . Let Q ∈ Γ( G ) with level ℓ . Then 2 is not a divisor of ℓ . W. Wang, A simple arithmetic criterion for graphs being determined by their generalized spectra, JCTB, 122 (2017) 438-451

The Theorem The Case p > 2 The Case p = 2 Examples Future Work Why p = 2 more difficult? Some important facts 1 Fact 1. e T A k e is always an integer for every integer k ≥ 1. 2 Fact 2. W T W ≡ 0 ( mod 2) (when n is odd, replace W with ¯ W , the matrix obtained from W be deleting the first row). 3 Fact 3. rank 2 ( W ) ≤ ⌈ n / 2 ⌉ . 4 Fact 4. Let det( W ) = ± 2 α p α 1 1 p α 2 2 · · · p α s s , then α ≥ [ n / 2]. 5 Fact 5. Let G ∈ F n . Then the SNF of W is diag (1 , 1 , · · · , 1 , 2 , 2 , · · · , 2 b ), where b is an odd square-free � �� � � �� � ⌈ n / 2 ⌉ ⌊ n / 2 ⌋ integer. 6 Fact 6. Let G ∈ F n . Then any ⌊ n 2 ⌋ linearly independent columns from W forms a set of fundamental solutions to the linear system of equation W T x = 0 over F 2 .

The Theorem The Case p > 2 The Case p = 2 Examples Future Work A Useful Lemma Lemma Let G ∈ G n and Q ∈ Γ( G ) with level ℓ . If 2 | ℓ , then there exists a (0 , 1) -vector u with u �≡ 0 ( mod 2) such that u T A k u ≡ 0 ( mod 4) , for k = 0 , 2 , · · · , n − 1 . (1) Moreover, u satisfies W T u ≡ 0 ( mod 2) .

The Theorem The Case p > 2 The Case p = 2 Examples Future Work A Key Lemma Lemma If 2 | ℓ , then the following linear system of equations has a non-trivial solution: W T ˜ W 1 x ≡ 0 (mod 2) . 2

The Theorem The Case p > 2 The Case p = 2 Examples Future Work Proof: A Sketch Proof. For simplicity, we just focus on the case n is even in what follows. The case n is odd can be proved similarly. Since W T W = ( e T A i + j − 2 e ) ≡ 0 (mod 2), there exists a Step 1: non-zero vector v such that u ≡ ˜ W v (mod 2), where ˜ W = [ e , Ae , · · · , A k − 1 e ], k = n 2 is fixed henceforth. Step 2: ∵ u = ˜ W v + 2 β ∴ u T A l u ≡ v T ˜ W T A l ˜ W v + 4 v T ˜ W T A l β + 4 β T A l β ≡ v T ˜ W T A l ˜ W v ≡ 0 (mod 4) , for any l ≥ 0.

The Theorem The Case p > 2 The Case p = 2 Examples Future Work Proof: A Sketch (continued) Proof. Step 3:   e T A l e e T A l +1 e e T A l + k − 1 e · · · e T A l +1 e e T A l +2 e e T A l + k e · · ·   W T A l ˜ ˜   W = . . . ...   . . . . . .   e T A l + k − 1 e e T A l + k e e T A l +2 k − 2 e , · · ·