  We study the spectral theory of a class of piecewise centrosymmetric Jacobi
operators defined on an associated family of substitution graphs. Given a
finite centrosymmetric matrix viewed as a weight matrix on a finite directed
path graph and a probabilistic Laplacian viewed as a weight matrix on a locally
finite strongly connected graph, we construct a new graph and a new operator by
edge substitution. Our main result proves that the spectral theory of the
piecewise centrosymmetric Jacobi operator can be explicitly related to the
spectral theory of the probabilistic Laplacian using certain orthogonal
polynomials. Our main tools involve the so-called spectral decimation, known
from the analysis on fractals, and the classical Schur complement. We include
several examples of self-similar Jacobi matrices that fit into our framework.
