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Eigenvalues and geometric representations of graphs László Lovász Microsoft Research One Microsoft Way, Redmond, WA 98052 [email protected] Matrices associated with graphs (all graphs connected) Adjacency matrix: Laplacian: A (aij ), aij # ij edges L (lij ) , aij , if i j , lij di , if i j. x T Lx ( xi x j ) 2 ij E Adjacency matrix: Laplacian: æ0 çç çç 1 çç çç 1 çç çç 0 çç è ç 1 æ çç çç çç çç çç çç çç è ç - 1 1 0 0 1 0 1 0 1 0 1 0 0 0 1 ö 1÷ ÷ ÷ 0÷ ÷ ÷ ÷ 0÷ ÷ ÷ ÷ 1÷ ÷ ÷ ÷ 0÷ ø ÷ ö 0 - 1÷ ÷ ÷ 1 2 - 1 0 0÷ ÷ ÷ ÷ 1 - 1 3 - 1 0÷ ÷ ÷ ÷ 0 0 - 1 2 - 1÷ ÷ ÷ ÷ ÷ 1 0 0 - 1 2ø ÷ 3 - 1 - 1 Eigenvalues and eigenvectors max 1 2 ... n min eigenvalues of adjacency matrix 1 2 ... n Tr( A) 0 min 0 1 2 ... n max eigenvalues of Laplacian Eigenvalues and eigenvectors 5 2 -1 1 0 1 -1 1 2.481... 4 0 1 1.481 0 1 -1 -1 1.193 1 1 1.481 The largest eigenvalue max max x T Ax x 2 daverage max dmax average degree dmax max dmax maximum degree If G is regular of degree d, then max d . The largest eigenvalue (G) max 1 (G) max 1 chromatic number Note: (G) (G) Wilf maximum clique 0 1 1 0 1 0 1 0 1 0 0 1 1 1 0 1 0 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 The largest eigenvalue 0 1 1 0 1 0 1 0 1 0 0 1 1 1 0 1 0 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 (G) max 1 not used! 1 7 0 1 1 .2 1 1 0 1 3 0 1 1 0 1 1.5 0 A ' .2 3 1 0 1 .2 0 1 1 0 1.5 1 0 7 1 0 .2 1 (G) max ( A ') 1 (G) min A' max ( A ') 1 (G) (G) The smallest eigenvalue G bipartite max min 1 1 1 -1 1 1 1 -1 1 1 1 -1 The smallest eigenvalue G k-colorable max (k 1)min max 1 (G) 1 max min (G) (G) (G) Hoffman Proof uses only 0 entries: can replace 1’s by anything maximizing we get: (G) Polynomial time computable! Computing (G) (G) min A' max ( A ') 1 1 7 0 1 1 .2 1 1 0 1 3 0 1 1 0 1 1.5 0 A ' .2 3 1 0 1 .2 0 1 1 0 1.5 1 0 .2 1 0 7 1 semidefinite optimization problem (G ) minimize t subject to X ii t 1 (i V ) X ij 1 (ij E ) ( X ij ) positive semidefinite Another matrix associated with graphs Adjacency matrix: Laplacian: Transition matrix (of random walk): A (aij ), aij # ij edges L (lij ) , aij , if i j , lij di , if i j. P ( pij ), pij aij di (Not much difference if graph is regular.) Random walks How long does it take to get completely lost? Sampling by random walk S: large and complicated set (all lattice random points inelement convex from body S Want: uniformly distributed Applications: - statistics all states of a physical system all matchings in a graph...) - simulation - counting - numerical integration - optimization - card shuffling... One general method for sampling: random walks (+rejection sampling, lifting,…) Want: sample from set V Construct regular connected non-bipartite graph with node set V Walk for T steps Output the final node ???????????? mixing time Example: random linear extension of partial order 5 4 5 4 2 3 3 2 1 1 Given: poset Node: compatible linear order Step: - pick randomly label i<n; - interchange i and i+1 if possible The second largest eigenvalue 1 1 1 2 ... n : eigenvalues of P A d max{2 , n } v0 , v1 ,..., vt ,...: random walk A Pr(v A) V t V t A V Conductance S V \S frequency of stepping from S to V \ S in random walk: E (S ,V \ S ) dV S V \S in sequence of independent samples: V 2 V E ( S ,V \ S ) ( S ) d S V \S Edge-density conductance: min S (S ) in cut Conductance and eigenvalue gap 1 1 2 ... n eigenvalues of transition matrix 2 1 2 Jerrum - Sinclair up to a constant factor Conductance and eigenvalue gap 1 1 2 ... n eigenvalues of transition matrix 2 1 2 Jerrum - Sinclair T x Ax T 2 max 2 : 1 x 0 x T T x Px T 1 x Lx T 1 2 min 1 : 1 x 0 min 2 : 1 x 0 2 x d x x T Lx ( xi x j ) 2 ij E What about the eigenvectors? 1 2 ... n eigenvalues of A G connected l1 has multiplicity 1 eigenvector is all-positive Frobenius-Perron What about the eigenvectors? 1 2 ... n eigenvalues of A Ax 2 x x0 x0 supp ( x), supp ( x) are connected. x0 Van der Holst