  We carry out a numerical and analytic analysis of the Yang-Lee zeros of the
1D Blume-Capel model with periodic boundary conditions and its generalization
on Feynman diagrams for which we include sums over all connected and
non-connected rings for a given number of spins. In both cases, for a specific
range of the parameters, the zeros originally on the unit circle are shown to
departure from it as we increase the temperature beyond some limit. The curve
of zeros can bifurcate and become two disjoint arcs as in the 2D case. We also
show that in the thermodynamic limit the zeros of both Blume-Capel models on
the static (connected ring) and on the dynamical (Feynman diagrams) lattice
tend to overlap. In the special case of the 1D Ising model on Feynman diagrams
we can prove for arbitrary number of spins that the Yang-Lee zeros must be on
the unit circle. The proof is based on a property of the zeros of Legendre
Polynomials.
