# The integrated density of states for the difference Laplacian on the modified Koch graph

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## Abstract

We consider the integrated density of states where

*N*(λ) of the difference Laplacian −Δ on the modified Koch graph. We show that*N*(λ) increases only with jumps and a set of jump points of*N*(λ) is the set of eigenvalues of −Δ with the infinite multiplicity. We establish also that$$0< C_1 \leqslant \mathop {\lim }\limits_{\lambda \to 0} \frac{{N(\lambda )}}{{\lambda ^{d_s /2} }}< \overline {\mathop {\lim }\limits_{\lambda \to 0} } \frac{{N(\lambda )}}{{\lambda ^{d_s /2} }} \leqslant C_2< \infty$$

*d*_{ s }=2log5/log(40/3) is the spectral dimension of MKG.## Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Spectral Dimension
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