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F is a linear operator in a N -dimensional vector space. We will investigate the spectral properties of this operator. Recall a few facts from Linear Algebra. Let A be a linear operator in a N -dimensional vector space V over R. A vector v 6= 0 is called an eigenvector of A if Av = v for some constant , that is called an eigenvalue of A. In general, one allows complex-valued eigenvalues by considering a complexi cation of V. The set of all complex eigenvalues of A is called the spectrum of A and is denoted by spec A.

5 For any nite connected weighted graph (V; ) of diameter D, c 1 D (V ) where c = minx y xy (for example, c = 1 for a simple weight). Proof. Let f be the eigenfunction of the eigenvalue 1. Let us normalize f to have max f = max jf j = 1; and let x0 be a vertex where f (x0 ) = 1. Since (f; 1) = 0, there is also a vertex y0 where f (y0 ) < 0. Let fxk gnk=0 be a path connecting x0 and y0 = xn where n D.

21) and cancelling by rk , we obtain the equation for r: r2 It has two complex roots r= where 2 r + 1 = 0: p i 1 2 =e 2 (0; ) is determined by the condition cos = (and sin = p i 1 ; 2 ). 21) f1 (k) = eik and f2 (k) = e ik : 48 CHAPTER 2. 22) are m-periodic provided m is a multiple of 2 , that is, 2 l ; m 2 (0; ) is equivalent to = for some integer l. The restriction l 2 (0; m=2) : Hence, for each l from this range we obtain an eigenvalue = cos of multiplicity 2 (with eigenfunctions cos k and sin k ).