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 Xi’an Jiaotong University Dec. 2016
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries Outline Introduction 1 Notations and Terminologies 2 A Simple Characterization of DGS Graphs 3 The Method 4 Examples 5 Summaries 6
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries Introduction 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 L. Babai (2015) There is a quasipolynomial time algorithm for all graphs, i.e., one with running time exp ((log n ) O (1) ). a a L. Babai, Graph isomorphism in quasipolynomial time, arXiv:1512.03547v2. W.X. Du (2016) The running time can be improved to n C log n for some constant C . a a W.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 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 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 A pair of cospectral graphs ✟✟✟✟✟✟ ❍❍❍❍❍❍ q q q q q q q q q q 0 1 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 spectrum: -2, 0, 0, 0, 2
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries Which Graphs Are DS? “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 Are almost all graphs DS? 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 Some Evidences Against The Conjecture Schwenk (1973) Almost every tree has a cospectral mate. a a A.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 Schwenk’s Constructions 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 respectively. Let Γ be a graph with root w. Then G • Γ and H • Γ are cospectral, where the new graphs are obtained from the old ones by identifying the roots u and w, and v with w.
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 The Conjecture Is False For Strongly Regular Graphs 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 Cospectral and Non-isomorphic Graphs Can Easily Be Constructed Godsil and McKay (1982), GM-switching Let the vertex set V of G can be partitioned into V and V 2 . Suppose G [ V 1 ] is regular, and every vertex in V 2 is adjacent to non, all, or exactly half number of vertices in V 1 . 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 Very Small Number of Graphs Are Known to Be DS 1 The complete graph K n . 2 The regular complete bipartite graph K n , n . 3 The cycle C n . 4 The path P n . 5 The tree Z n . 6 · · · · · · · · ·
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries Computer Enumerations 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 Our Results on Generalized Spectral Characterization of Graphs 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 Notations and Terminologies 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 P G 1 ( λ ) = P G 2 ( λ ) = λ 7 − 6 λ 5 + 9 λ 3 − 4 λ G 2 ( λ ) = λ 7 − 15 λ 5 − 2 λ 4 + 12 λ 3 + 24 λ 2 P ¯ G 1 ( λ ) = P ¯
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries DGS Graphs 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 DGS Graphs: The basics The walk-matrix of graph G: W ( G ) = [ e , A ( G ) e , . . . , A ( G ) n − 1 e ] 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 Controllable Graphs 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 G n . It was conjectured (by C.D. Godsil) that almost all graphs are controllable. ′ Rourke and Touri showed recently that this conjecture is O true.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries A Simple Characterization Theorem 1. [Wang and Xu, 2006] Let G ∈ G n . 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 Q T A ( G ) Q = A ( H ) , and Qe = e , (1) where e is the all-ones vector. a a 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.
Introduction Notations and Terminologies A Simple Characterization of DGS Graphs The Method Examples Summaries The Proof: Some Lemmas Lemma Let G be a graph with adjacency matrix A. Let A k = ( a ( k ) ij ) . Then a ( k ) equals the number of walks of length k in G starting from i ij and ending at j. Lemma Let N k ( G ) be the number of walks of length k in G. Then N k ( G ) = e T Ae.
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