Download PDF by Alexander Grigoryan: Analysis on Graphs

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1 Cheeger's inequality Let (V; ) be a weighted graph with the edges set E. Recall that, for any vertex subset V , its measure ( ) is de ned by X ( )= (x) : x2 Similarly, for any edge subset S E, de ne its measure X (S) = ; (S) by 2S where := xy for any edge = xy: For any set V , de ne its edge boundary @ @ by = g: = fxy 2 E : x 2 ; y 2 De nition. 1) In other words, h is the largest constant such that the following inequality is true (@ ) for any subset of V with measure ( ) h ( ) 1 2 57 (V ). 2) 58 CHAPTER 3.

11)). 9) follows. 8). We have seen that a random walk on a nite, connected, non-bipartite graph is ergodic. Let us show that if N 1 = 2 then this is not the case (as we will see later, for bipartite graphs one has exactly N 1 = 2): Indeed, if f is an eigenfunction of L with the eigenvalue 2 then f is the eigenfunction of P with the eigenvalue 1, that is, P f = f . Then we obtain that P n f = ( 1)n f so that P n f does not converge to any function as n ! 1. 12) 42 CHAPTER 2. SPECTRAL PROPERTIES OF THE LAPLACE OPERATOR The value of " should be chosen so that s (x) " << (x0 ) (x) ; (V ) which is equivalent to (x) : (V ) " << min x In many examples of large graphs, 1 is close to 0 and N 1 is close to 2.

2. Let V = f1; 2; 3g with edges 1 2 then Lf (1) = f (1) 3 1, that is, (V; E) = C3 = K3 . We have 1 (f (2) + f (3)) 2 and similar identities for Lf (2) and Lf (3) : The action of L can be written as a matrix multiplication: 0 1 0 10 1 Lf (1) 1 1=2 1=2 f (1) @ Lf (2) A = @ 1=2 1 1=2 A @ f (2) A : Lf (3) 1=2 1=2 1 f (3) 3 The characteristic polynomial of the above 3 3 matrix is 3 2 + 49 its roots, we obtain the following eigenvalues of L: = 0 (simple) and multiplicity 2. 3. Let V = f1; 2; 3g with edges 1 2 3.

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Analysis on Graphs by Alexander Grigoryan

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