On abelian $$\ell $$-towers of multigraphs II

Let $$\ell $$ be a rational prime. Previously, abelian -towers of multigraphs were introduced which are analogous to $${\mathbb {Z}}_{\ell }$$ -extensions number fields. It was shown that for certain class towers bouquets, the growth -part spanning trees behaves in predictable manner (analogous well-known theorem Iwasawa fields). In this paper, we give generalization broader regular bouquets than originally considered. To carry out, observe shifted Chebyshev polynomials members continuously parametrized family power series with coefficients and then study special value at $$u=1$$ Artin-Ihara L-function -adically.