Cospectral Graphs 1-Full Graphs Are Almost All Graphs Cospectral? Chris Godsil November 9, 2007 Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Outline 1 Cospectral Graphs Polynomials and Walks Constructing Cospectral Graphs Switching 2 1-Full Graphs A Cyclic Subspace Generating All Matrices Cospectral 1-Full Graphs Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Polynomials and Walks Outline 1 Cospectral Graphs Polynomials and Walks Constructing Cospectral Graphs Switching 2 1-Full Graphs A Cyclic Subspace Generating All Matrices Cospectral 1-Full Graphs Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Polynomials and Walks The Characteristic Polynomial Definition Let G be a graph with adjacency matrix A. The characteristic polynomial φ(G; t) of G is the characteristic polynomial of A: φ(G; t) := det(tI − A): Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Polynomials and Walks Examples The characteristic polynomials of K1, K2 and P3 are respectively: t; t2 − 1; t3 − 2t: Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Polynomials and Walks Walks Lemma r If A is the adjacency matrix of G, then (A )i;j is the number of walks in G from vertex i to vertex j with length r. Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Polynomials and Walks Closed Walks A walk in G is closed if its first and last vertices are equal. The number of closed walks in G with length r is X r r (A )i;i = tr(A ): i2V (G) Chris Godsil Are Almost All Graphs Cospectral? (And then it gets messy!) Cospectral Graphs 1-Full Graphs Polynomials and Walks Short Closed Walks If G has n vertices, e edges and contains exactly t triangles, then tr(A0) = n tr(A1) = 0 tr(A2) = 2e tr(A3) = 6t: Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Polynomials and Walks Short Closed Walks If G has n vertices, e edges and contains exactly t triangles, then tr(A0) = n tr(A1) = 0 tr(A2) = 2e tr(A3) = 6t: (And then it gets messy!) Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Polynomials and Walks A Generating Function The generating function for the closed walks in G, counted by length, is X tr(Ar)tr: r≥0 It is a rational function: X t−1φ0(G; t−1) tr(Ar)tr = : φ(G; t−1) r≥0 Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Polynomials and Walks A Characterisation Corollary Two graphs G and H are cospectral if and only if their generating functions for closed walks are equal. Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Polynomials and Walks The Smallest Cospectral Graphs Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Polynomials and Walks The Smallest Connected Cospectral Graphs Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Polynomials and Walks Cospectral, Cospectral Complements? The graphs C4 [ K1 and K1;4 have two and four triangles respectively|they are not cospectral. Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Polynomials and Walks Another Generating Function The number of walks of length r in G is equal to tr(ArJ) = 1T Ar1 and thus X tr(ArJ)tr r≥0 is the generating function for all walks in G. Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Polynomials and Walks Complements and Walks Theorem Suppose G and H are cospectral graphs with respective adjacency matrices A and B. Then G and H are cospectral if and only if the generating functions for all walks in G and in H are equal. Chris Godsil Are Almost All Graphs Cospectral? Lemma Cospectral regular graphs have cospectral complements. Cospectral Graphs 1-Full Graphs Polynomials and Walks Regular Graphs If G is a k-regular graph on n vertices then its walk generating function is n : 1 − kt Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Polynomials and Walks Regular Graphs If G is a k-regular graph on n vertices then its walk generating function is n : 1 − kt Lemma Cospectral regular graphs have cospectral complements. Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Polynomials and Walks Two Irregular Graphs Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Constructing Cospectral Graphs Outline 1 Cospectral Graphs Polynomials and Walks Constructing Cospectral Graphs Switching 2 1-Full Graphs A Cyclic Subspace Generating All Matrices Cospectral 1-Full Graphs Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Constructing Cospectral Graphs 0-Sums The 0-sum of two graphs G and H is got by identifying a vertex in G with a vertex in H: G v H Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Constructing Cospectral Graphs Spectrum of a 0-Sum If we create the 0-sum F by merging v in G with v in H, then φ(F ) = φ(G)φ(H n v) + φ(G n v)φ(H) − tφ(G n v)φ(H n v): Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Constructing Cospectral Graphs Example If G = K2 and K = K2 then their 0-sum F is P3, whence 2 2 2 3 φ(P3; t) = (t − 1)t + t(t − 1) − t(t ) = t − 2t: Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Constructing Cospectral Graphs Corollary If we hold G and its vertex v fixed, then the characteristic polynomial of the 0-sum of G and H is determined by the characteristic polynomials of H and H n v. Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Constructing Cospectral Graphs Deleting Vertices If H is the graph u v then H n u and H n v are isomorphic. Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Constructing Cospectral Graphs A Cospectral Pair . and thus we obtain a pair of cospectral graphs: G G Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Constructing Cospectral Graphs Another Pair G G Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Constructing Cospectral Graphs A Hint u v Chris Godsil Are Almost All Graphs Cospectral? with cospectral complements. Cospectral Graphs 1-Full Graphs Constructing Cospectral Graphs A Theorem Theorem (Schwenk. Godsil & McKay) Almost all trees are cospectral. Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Constructing Cospectral Graphs A Theorem Theorem (Schwenk. Godsil & McKay) Almost all trees are cospectral. with cospectral complements. Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Switching Outline 1 Cospectral Graphs Polynomials and Walks Constructing Cospectral Graphs Switching 2 1-Full Graphs A Cyclic Subspace Generating All Matrices Cospectral 1-Full Graphs Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Switching Yet Another Construction G G Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Switching If 0−1 1 1 1 1 01 1 11 1 B 1 −1 1 1 C B0 1 0C K := B C ;M = B C 2 @ 1 1 −1 1 A @0 0 1A 1 1 1 −1 1 0 0 then K1 = 1 and K2 = I, whence K is orthogonal, and 00 0 01 B1 0 1C KM = B C = J4 − M: @1 1 0A 0 1 1 Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Switching 0 1 0 1 0 1 K 0 0 0 M 0 K 0 0 B C B T C B C B 0 I 0C BM A1 B1C B 0 I 0C @ A @ A @ A T 0 0 I 0 B1 A2 0 0 I 0 0 J − M 0 1 T = @(J − M) A1 B1A T 0 B1 A2 Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Switching Therefore. Theorem Switching related graphs are cospectral, with cospectral complements. Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Switching A Problem Is it true that almost all graphs are determined by their spectrum? Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs Switching A Related Example H H Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs A Cyclic Subspace Outline 1 Cospectral Graphs Polynomials and Walks Constructing Cospectral Graphs Switching 2 1-Full Graphs A Cyclic Subspace Generating All Matrices Cospectral 1-Full Graphs Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs A Cyclic Subspace A Subspace Let G be a graph on n vertices with adjacency matrix A. Define U n r to be the subspace of R spanned by the vectors A 1, for all non-negative integers r. Chris Godsil Are Almost All Graphs Cospectral? Proof. If P is a permutation matrix, P 1 = 1. If P 2 Aut(G), then PA = AP and so, for all r PAr1 = ArP 1 = Ar1: Cospectral Graphs 1-Full Graphs A Cyclic Subspace Automorphisms Theorem If the permutation matrix P is in Aut(G), then P u = u for all u in U. Chris Godsil Are Almost All Graphs Cospectral? Cospectral Graphs 1-Full Graphs A Cyclic Subspace Automorphisms Theorem If the permutation matrix P is in Aut(G), then P u = u for all u in U.