Some results concerning the $p$-Royden and $p$-harmonic boundaries of a graph of bounded degree

Let $p$ be a real number greater than one and let $\mathrm{\Gamma }$ be a connected graph of bounded degree. We show that the $p$-Royden boundary of $\mathrm{\Gamma }$ with the $p$-harmonic boundary removed is a $F_{\sigma }$-set. We also characterize the $p$-harmonic boundary of $\mathrm{\Gamma }$ in terms of the intersection of the extreme points of a certain subset of one-sided infinite paths in $\mathrm{\Gamma }$.