The definition of Markov operator which I am familiar with:
For a graph $ G=(V,E)$ , Markov’s operator upon a function $ \varphi:V\rightarrow\mathbb{C}$ , $ \varphi\in L^2(G,\nu)$ ($ \nu(v):=\deg(v)$ ) is defined in the following manner: $ M\varphi(x) = \frac{1}{\deg(x)}\sum_{y\in N(x)} \varphi(y)$ , when $ N(x)$ denotes $ x$ ‘s neighborhood in $ G$ .
As part of expander graphs studies, I am searching for some articles or books for known results about the spectral radius of Markov’s operator, and the spectral radius of the operator when it is defined on $ L^2(G,\nu)\setminus\{\mathrm{ker}(M-ID)\uplus \mathrm{ker}(M+ID)\}$ , which is used to evaluate Cheeger’s constant. (I am working now with “Introduction to expander graphs” by E. Kowalski, which shows some results for complete graphs, circle and the famous result by Kesten of $ T_d$ , $ d$ -regular tree).
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