Generalized Spectral Characterization of Graphs: Revisited

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

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, Fp denotes a finite field with p elements, denotes the rank of W over Fp.

8. DGS Graphs • Two graphs are cospectralw.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 rationalorthogonal 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 Fp . • If over Fp , 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 Fp , . • 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 Fp , . • 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!