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We investigate the method of moments for $d$-regular digraphs and thelimiting $d$-regular directed tree $T_d$ as the number of vertices tends toinfinity, in the same spirit as McKay (Linear Algebra Appl., 1981) for theundirected setting. In particular, we provide a combinatorial derivation of theformula for the star moments (from a root vertex $o\in T_d$)$$M_d(w)\qquad:=\sum_{\substack{v_0,v_1\ldots,v_{k-1},v_k\in T_d\\v_0=v_k=o}}A^{w_1}(v_0,v_1)A^{w_2}(v_1,v_2) \cdots A^{w_k}(v_{k-1},v_k)$$ with $A$ theadjacency matrix of $T_d$, where $w:=w_1\cdots w_k$ is any word on the alphabet$\{1,{*}\}$ and $A^*$ is the adjoint matrix of $A$. Our analysis highlights aconnection between the non-zero summands of $M_d(w)$ and the non-crossingpartitions of $\{1,\ldots,k\}$ which are in some sense compatible with $w$.

A combinatorial view on star moments of regular directed graphs and trees