  We establish asymptotics of growing one dimensional self-similar fractal
graphs, they are networks that allow multiple weighted edges between nodes, in
terms of quantum central limit theorems for algebraic probability spaces in
pure state. An additional structure is endowed with the repeating units of
centro-symmetric Jacobians in the adjacency of a linear graph creating a
self-similar fractal. The family of fractals induced by centro-symmetric
Jacobians formulated as orthogonal polynomials that satisfy three term
recurrence relations support such limits. The construction proceeds with the
interacting fock spaces, T-algebras endowed with a quantum probability space,
corresponding to the Jacobi coefficients of the recurrence relations and when
some elements of the centro-symmetric matrix are constrained in a specific way
we obtain, as the same Jacobian structure is repeated, the central limits. The
generic formulation of Leonard pairs that form bases of conformal blocks and
probablistic laplacians used in physics provide choice of centro-symmetric
Jacobians widening the applicability of the result. We establish that the
T-algebras of these 1D fractals, as they form a special class of
distance-regular graphs, are thin and the induced association schemes are
self-duals that lead to anyon systems with modular invariance.
