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We study the Wold decomposition for representations of a self-similarsemigroup $P$ action on a directed graph $E$. We then apply this decompositionto the case where $P=\bN$ to study the C*-envelope of the associated universalnon-selfadjoint operator algebra $\A_{\bN, E}$ by carefully constructingexplicit non-trivial dilations for non-boundary representations. In particular,it is shown that the C*-envelope of $\A_{\bN, E}$ coincides with theself-similar C*-algebra $\O_{\bZ, E}$.

Wold Decomposition and C*-envelopes of Self-Similar Semigroup Actions on Graphs