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By Alexander Grigoryan

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Adding together the last two lines and dividing by 2, we obtain X f (x)g(x) (x) = x2 1 X (f (y) 2 x;y2 f (x)) (g(y) g(x)) xy + XX x2 y2 (rxy f ) g(x) xy ; c which was to be proved. 2 Eigenvalues of the Laplace operator Let (V; ) be a nite connected weighted graph where N := #V > 1: Let F denote the set of real-valued functions on V . Then F is a vector space over R of dimension N . Hence, the Laplace operator : F ! F is a linear operator in a N -dimensional vector space. We will investigate the spectral properties of this operator.

5 for bipartite graphs (see Exercise 20 for proofs). 7 Let (V; ) be a nite connected weighted graph. Assume that (V; ) is bipartite, and let V + ; V be a bipartition of V: For any function f on V , consider the function fe on V that takes two values as follows: P 2 f (y) (y) ; x 2 V + ; e Py2V + f (x) = (V ) y2V f (y) (y) ; x 2 V : Then, for all even n, P nf where = max (j1 fe n kf k 1j ; j N 2 1j) : Consequently, for all x 2 V , we have P n f (x) ! fe(x) as n ! 1, n is even. Note that 0 hence, 0 < 1 N < 1 because the eigenvalues 2 < 2.

SPECTRAL PROPERTIES OF THE LAPLACE OPERATOR q , p+q respectively, and then chooses a vertex in the chosen direction accordingly to the Markov kernel there. In particular, if a and b are simple weights, then we obtain 8 < p deg (y) ; if x x0 and y = y 0 ; q deg (x) ; if y y 0 ; and x = x0 ; (x;y);(x0 ;y 0 ) = : 0; otherwise: If in addition the graphs A and B are regular, that is, deg (x) = const =: deg (A) and deg (y) = const =: deg (B) then the most natural choice of the parameter p and q is as follows 1 1 and q = ; p= deg (B) deg (A) so that the weight is also simple.

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