A recent example of a non-hyponormal injective composition operator in an$L^2$-space generating Stieltjes moment sequences, invented by three of thepresent authors, was built over a non-locally finite directed tree. The maingoal of this paper is to solve the problem of whether there exists such anoperator over a locally finite directed graph and, in the affirmative case, tofind the simplest possible graph with these properties (simplicity refers tolocal valency). The problem is solved affirmatively for the locally finitedirected graph $\mathscr G_{2,0}$, which consists of two branches and one loop.The only simpler directed graph for which the problem remains unsolved consistsof one branch and one loop. The consistency condition, the only efficient toolfor verifying subnormality of unbounded composition operators, is intensivelystudied in the context of $\mathscr G_{2,0}$, which leads to a constructivemethod of solving the problem. The method itself is partly based ontransforming the Krein and the Friedrichs measures coming either from shiftedAl-Salam-Carlitz $q$-polynomials or from a quartic birth and death process.
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