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

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