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 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). 



Spectral radius of Markov averaging operator on graphs
The Markov averaging operator $M$ is also known as the transition matrix for simple random walk on the graph (often denoted by $P$). With this terminology, its spectral radius and spectral gap are studied extensively; convenient starting points are the lecture notes [1] and Chapters 12-13 in the book [2] and the references therein (See also Chapter 19 in [2] for a nice class of examples). Chapter 6 in [3] is also relevant. On finite graphs, the spectral radius of $M$ on $L^2(\nu)$ is 1, so one often considers the second largest eigenvalue or the second largest eigenvalue in absolute value, which is equivalent to the spectral radius of $M$ restricted to a subspace as in your query. For infinite graphs, Woess' book [4] is a useful reference, and Chapter 6 in [3] also discusses it. For the case of Cayley graphs of infinite groups see [4] or Chapter 14 in [3].


[1] Saloff-Coste, L. "Lectures on finite Markov chains, volume 1665. Ecole d’Eté de Probabilités de Saint Flour, P. Bernard, ed." Lecture Notes in Mathematics, Springer, Berlin (1997).


[2] Markov chains and mixing times, by David A. Levin and Yuval Peres, with contributions by Elizabeth L. Wilmer. American Mathematical Society, 2017 Available at <https://pages.uoregon.edu/dlevin/MARKOV/mcmt2e.pdf>


[3] Lyons, Russell, and Yuval Peres. Probability on trees and networks. Cambridge University Press, 2017. Available at <http://mypage.iu.edu/~rdlyons/prbtree/prbtree.html>


[4] Woess, Wolfgang. Random walks on infinite graphs and groups. Vol. 138. Cambridge university press, 2000.


