By Alexander Grigoryan

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Here " must be chosen so that " << minx Eigenvalues of Zm Let us give an example of computation of the eigenvalues k of L: Frequently it is more convenient to compute the eigenvalues k = 1 k of the Markov operator P = id L: Let us compute the eigenvalues of the Markov operator on the cycle graph Cm with simple weight. Recall that Cm = Zm = f0; 1; :::; m 1g and the connections are 0 1 2 ::: m 1 0: The Markov operator is given by P f (k) = 1 (f (k + 1) + f (k 2 1)) where k is regarded as a residue mod m.

Chapter 2 Spectral properties of the Laplace operator Let (V; ) be a locally nite weighted graph without isolated points and weighted Laplace operator on (V; ). 1 be the Green's formula Let us consider the di erence operator rxy that is de ned for any two vertices x; y 2 V and maps F to R as follows: rxy f = f (y) f (x) : The relation between the Laplace operator f (x) = and the di erence operator is given by 1 X (rxy f ) (x) y xy = X P (x; y) (rxy f ) y Indeed, the right hand side here is equal to X X f (y) P (x; y) (f (y) f (x)) P (x; y) = y y = X f (y) P (x; y) X f (x) P (x; y) y f (x) = f (x) : y The following theorem is one of the main tools when working with the Laplace operator.

The eigenvectors of are also called the eigenfunctions. 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 ).