Let $ u_0 geq u_1 geq cdots geq u_ {n-1} $ be positive numbers and define a matrix $ n times n $ by $ M_ {i, j} = u _ { left | i-j right |} $ for everyone $ i, j $.

Let $ L = D – M $ to be the Laplacian matrix of $ M $, or $ D $ is a diagonal matrix such that $ D_ {ii} = sum_j M_ {i, j} $ for everyone $ i $. Yes $ lambda_1 = 0 leq lambda_2 leq cdots leq lambda_n $ are the eigenvalues of $ L $. Then we can show that $ lambda_2 $ has a monotone clean vector, called Fiedler vector.

(for proof see eg https://arxiv.org/abs/1411.0210).

If we put $ L = I – D ^ {- 1/2} MD ^ {- 1/2} $ is it possible to get the same result? It is that there is a monotone clean vector $ v $ corresponding to the second smallest eigenvalue of $ L & # 39;?