# mutual coherence

I read Vertex-Frequency Analysis on Graphs (arxiv link)

Shuman, David I, et al. “Vertex-Frequency Analysis on Graphs.” Applied and Computational Harmonic Analysis, vol. 40, no. 2, 2016, pp. 260–291., doi:10.1016/j.acha.2015.02.005.

there are many articles about the localization properties of graph Laplacian eigenvectors. For different classes of random graphs, show that with high probability for graphs of sufficiently large size, the eigenvectors of the graph Laplacian (or in some cases, the graph adjacency operator), are delocalized; i.e., the restriction of the eigenvector to a large set must have substantial energy, or in even stronger statements, the element of the matrix $$\chi:=[\chi_0,\chi_1,...,\chi_{N−1}]$$ with the largest absolute value is small. We refer to this latter value as the mutual coherence (or simply coherence) between the basis of Kronecker deltas on the graph and the basis of graph Laplacian eigenvectors:

$$\mu := \max_{\scriptscriptstyle 0\le \ell\le N-1\atop {1\le i\le N}} = \vert \langle \chi_\ell ,\delta_i \rangle \vert \in \Big[ 1/\sqrt{N} , 1\Big]$$.

• I want to understand that What's the intuition behind $$\mu$$.
• What is $$\mu$$ exactly ?
• what is coherence in the graph setting?
• too, what is mutual coherence in the graph setting?

(I don 't know localized eigenvector and delocalized eigenvectors properties,i.e. I have a lot of information about random graphs ).

• This is probably "mutual", not "mudual". – Helen Sep 30 '18 at 12:02
• @Helen yes, sorry, Edited. – niloofar jamshidi Sep 30 '18 at 12:06
• If you edit it in the title as well, it will help attract more readers :) – Helen Sep 30 '18 at 12:26