  This paper deals with the location of the complex zeros of the Tutte
polynomial for a class of self-dual graphs. For this class of graphs, as the
form of the eigenvalues is known, the regions of the complex plane can be
focused on the sets where there is only one dominant eigenvalue in particular
containing the positive half plane. Thus, in these regions, the analyticity of
the pressure can be derived easily. Next, some examples of graphs with their
Tutte polynomial having a few number of eigenvalues are given. The cases of the
strip of triangles with a double edge, the wheel and the cycle with an edge
having a high order of multiplicity are presented. In particular, for this last
example, we remark that the well known conjecture of Chen et al. is false in
the finite case.
