On the spectrum of the averaging operator for a finite homogeneous graph

An estimation of the spectrum of the averaging operator $T_i(\Gamma,1)$ over the radius 1 for a finite $(q+1)$-homogeneous quotient graph $\Gamma\setminus X$, where $X$ is an infinite $(q+1)$-homogeneous tree associated with free group $G$ on the finite set of generators $S=\{x_1,\dots,x_p\}$ ($2p=q+1$), $\Gamma$ is a subgroup in $G$ of finite index, in the subspace $L^2(\Gamma\setminus G,1)\ominus E_{ex}$ where $E_{ex}$ is a subspace of eigenfunctions of $T_1(\Gamma,1)$ with eigenvalue $\lambda$ such that $|\lambda|=q+1$, is given. A construction of some finite homogeneous graphs is presented, for which the spectrum of their adjacency matrices can be calculated explicitely. Bibliography: 11 titles.