Graph power and walk constraint

For a $d$-regular constraint graph with normalized adjacency $M$ under Constraint graph and labeling value, and an integer $t\ge 1$, the adjacency-slot graph power has one slot for every length-$t$ port walk. Its degree is $d^t$, its adjacency is $A^t$, and its transition matrix is $M^t$: matrix multiplication counts walks with all multiplicities. Walk reversal supplies an inverse slot.

The predicate of a walk $(v_0,e_1,v_1,\dots ,e_t,v_t)$ tests every original edge relation on the labels of its incident vertex occurrences. Repeated occurrences of one vertex use the same label. Thus it is a conjunction of $t$ tests, on at most $t+1$ original vertices; it is not in general a binary predicate on endpoint letters. A fixed labeling satisfies the walk predicate exactly when none of the traversed edges is violated.