  We prove several theorems concerning Tutte polynomials $T(G,x,y)$ for
recursive families of graphs. In addition to its interest in mathematics, the
Tutte polynomial is equivalent to an important function in statistical physics,
the Potts model partition function of the $q$-state Potts model, $Z(G,q,v)$,
where $v$ is a temperature-dependent variable. We determine the structure of
the Tutte polynomial for a cyclic clan graph $G[(K_r)_m,L=jn]$ comprised of a
chain of $m$ copies of the complete graph $K_r$ such that the linkage $L$
between each successive pair of $K_r$'s is a join $jn$, and $r$ and $m$ are
arbitrary. The explicit calculation of the case $r=3$ (for arbitrary $m$) is
presented. The continuous accumulation set of the zeros of $Z$ in the limit $m
\to \infty$ is considered. Further, we present calculations of two special
cases of Tutte polynomials, namely, flow and reliability polynomials, for
cyclic clan graphs and discuss the respective continuous accumulation sets of
their zeros in the limit $m \to \infty$. Special valuations of Tutte
polynomials give enumerations of spanning trees and acyclic orientations. Two
theorems are presented that determine the number of spanning trees on
$G[(K_r)_m,jn]$ and $G[(K_r)_m,id]$, where $L=id$ means that the identity
linkage. We report calculations of the number of acyclic orientations for
strips of the square lattice and use these to obtain an improved lower bound on
the exponential growth rate of the number of these acyclic orientations.
