1. Wei Wang Xi’an Jiaotong University Generalized Spectral Characterization of Graphs: Revisited Shanghai Conference on Algebraic Combinatorics (SCAC), Shanghai, Aug, 2012

2. Outline Introduction Review of Some Old Results Some New Results Summary An Open Problem for Further Research

3. Introduction 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 The Laplacian spectrum, one can deduce : (i) the number of spanning trees; (ii) the number of connected components; ……………. Question: Can graphs be determined by the spectrum?

4. Cospectral Graphs A pair of cospectral graphs; Schwenk (1973): Almost no trees are determined by the spectrum.

5. DS Graphs Question: Which graphs are determined by their spectrum (DS for short) ? This is an old unsolved problem in Spectral Graph Theory that dates back to more than 50 years. Applications: Chemistry; Graph Isomorphism Problem; The shape and sound of a drum (“Can one hear the shape of the drum?”); ……

6. Two recent survey papers : E. R. van Dam, W. H. Haemers, Which graphs are determined by their spectrum? Linear Algebra Appl. 373 (2003) 241-272. E. R. van Dam, W. H. Haemers, Developments on spectral characterizations of graphs, Discrete Mathematics, 309 (2009) 576-586. This talk will focus on the topic of characterizing graphs by both the spectrum and the spectrum of the complemnt of a graph.

7. Notations and Terminologies : a simple graph (unless stated otherwise) vertex set ; edge set. The adjacency matrix of graph G is an matrix with. The characteristic polynomial of graph G: The spectrum of G is the multiset of all the eigenvalues of Two graphs are cospectral if. denotes a prime number, F p denotes a finite field with p elements, denotes the rank of W over F p.

8. DGS Graphs Two graphs are cospectral w.r.t. the generalized spectrum if and. A graph 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.

9. DGS Graphs: An Review of Some Old Results The walk-matrix of graph G: where is the all-one vector. Remarks: 1. The -th entry of W is the number of all walks starting from vertex with length. 2. The arithmetic properties of det(W) is crucial for our discussions.

10. Controllable Graph A graph G is called a controllable graph if the corresponding walk-matrix is non-singular. The set of all controllable graphs of order is denoted by

11. A Simple Characterization Theorem 1. [Wang and Xu,2006] Let. Then there exists a graph H with and if and only if there exists a unique rational orthogonal matrix Q such that (1)

12. A Simple Characterization Define. Theorem 2. [Wang and Xu, 2006] Let. Then G is DGS if and only if contains only permutation matrices. Question: How to find out all ?

13. The Level of Q Definition: Let Q be a rational orthogonal matrix with Qe=e, the level of Q is the smallest positive integer such that is an integral matrix. If, then Q is a permutation matrix. Example:

14. The Smith Normal Form An integral matrix is called unimodular if. Let be an n by n integral matrix with full rank. Then there exist two unimodular matrices and such that where is called the i-th elenmentry divisor of.

15. An Exclusion Principle Lemma. [Wang and Xu,2006] Let W be the walk-matrix of a graph.Let Then we have

16. Some Basic Ideas (i) All the possible prime divisors of is finite; they are the divisors of, and hence are divisors of. (ii) Some of the prime divisors of may not be divisors of, they can be excluded from further consideration. (iii) If all the prime factors of are not divisors of,, then we must have =1, and hence contains only permutation matrices and G is DGS.

17. Primes p>2 Let be a prime,. If Eq. (1) has no solution, then is not a divisor of. Assume, the solution to the system of linear equations can be written as over finite field F p. If over F p, then is not a divisor of. Using this way, the odd prime divisors of can be excluded in most cases.

18. Examples

19. The First Graph It can be computed. For p=17,67,8054231, solve Eq (1) and check whether is zero or not over F p,. All primes (except p=2) can be excluded.

20. The Second Graph It can be computed. For p=3,5,197,263,5821, solve Eq (1) and check whether is zero or not over F p,. All primes (except p=2,5) can be excluded.

21. The prime p=2 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 have to develop more intensive exclusion conditions. I shall not go into the details. To conclude, it can be shown that the first graph is DGS. But cannot be shown to be DGS, since p=5 cannot be excluded by using the existing methods.

22. Question: Does there exist a simple method to exclude the primes p>2? (The case p=2 is more involved, I shall concentrate on the case p>2 in this talk.)

23. A New Exclusion Principle for p>2 Theorem 3. [Wang,2012] Let. Let Suppose that, where is an odd prime. Then is not a divisor of.

24. A diagram for the Proof of Theorem 3

25. The Proof: A sketch

28. Example and counterexample In previous example, is DGS, since p=5 can be excluded by Theorem 3. Theorem 3 may be false if the exponents of p>2 is larger than 1. Let the adjacency matrix of G be given as follows, is a (0,1)matrix, and hence is an adjacency matrix of another graph H. However, note that Thus, p=3 cannot be excluded.

29. An Counter-example.

30. Summary We have reviewed some existing methods for showing a graph to be DGS; in particular, we review the exclusion principle for excluding those odd primes of det(W). We also present a simple new criterion to exclude all odd prime factors with exponents one in the prime decomposition of det(W); It suggests that the arithmetic properties of det(W) contains much information about whether G is DGS or not.

31. Problem for Further Research Conjecture [Wang,2006]. Let. Then G is DGS if (which is always an integer) is square-free. Remarks: i) We have shown that odd primes p>2 with exponents one can be excluded. ii) The case p=2 still needs further investigations!!!

32. References [1] E. R. van Dam, W. H. Haemers, Which graphs are determined by their spectrum? Linear Algebra Appl., 373 (2003) 241-272. [2] E. R. van Dam, W. H. Haemers, Developments on spectral characterizations of graphs, Discrete Mathematics, 309 (2009) 576-586. [3] 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. [4] 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. [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, manuscript, Aug, 2012.

33. Thank you! The end!