By Alexander Grigoryan
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5). Now assume that the weights mk satisfy a stronger condition mk+1 cmk ; 1 is 60 CHAPTER 3. 6) Let us estimate the mixing time on the above path graph (V; ). 1. CHEEGER'S INEQUALITY 61 Consider the weights mk = ck where c > 1. 6), we obtain T c+1 c 1 4N ln c 2 : Note that T is linear in N ! Consider one more example: mk = k p for some p > 1. 2 by two lemmas. Given a function f : V ! 3 (Co-area formula). Given any real-valued function f on V , set for any t 2 R t = fx 2 V : f (x) > tg: Then the following identity holds: X 2E 1 jr f j Note that r f is unde ned unless the edge = Z 1 (@ t ) dt: 1 is directed, whereas jr f j makes always sense.
2. EIGENVALUES OF THE LAPLACE OPERATOR 35 which can be considered as the integration of f g against measure on V . Obviously, all axioms of an inner product are satis ed: (f; g) is bilinear, symmetric, and positive de nite (the latter means that (f; f ) > 0 for all f 6= 0). 2 The operator is, is symmetric with respect to the above inner product, that ( f; g) = (f; g) for all f; g 2 F. Proof. 2), we have ( f; g) = X f (x) g (x) (x) = x2V 1 X (rxy f ) (rxy g) 2 x;y2V xy ; and the last expression is symmetric in f; g so that it is equal also to ( g; f ).
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.
Analysis on Graphs by Alexander Grigoryan