  We consider the $q$-state Potts model on families of self-dual strip graphs
$G_D$ of the square lattice of width $L_y$ and arbitrarily great length $L_x$,
with periodic longitudinal boundary conditions. The general partition function
$Z$ and the T=0 antiferromagnetic special case $P$ (chromatic polynomial) have
the respective forms $\sum_{j=1}^{N_{F,L_y,\lambda}} c_{F,L_y,j}
(\lambda_{F,L_y,j})^{L_x}$, with $F=Z,P$. For arbitrary $L_y$, we determine (i)
the general coefficient $c_{F,L_y,j}$ in terms of Chebyshev polynomials, (ii)
the number $n_F(L_y,d)$ of terms with each type of coefficient, and (iii) the
total number of terms $N_{F,L_y,\lambda}$. We point out interesting connections
between the $n_Z(L_y,d)$ and Temperley-Lieb algebras, and between the
$N_{F,L_y,\lambda}$ and enumerations of directed lattice animals. Exact
calculations of $P$ are presented for $2 \le L_y \le 4$. In the limit of
infinite length, we calculate the ground state degeneracy per site (exponent of
the ground state entropy), $W(q)$. Generalizing $q$ from ${\mathbb Z}_+$ to
${\mathbb C}$, we determine the continuous locus ${\cal B}$ in the complex $q$
plane where $W(q)$ is singular. We find the interesting result that for all
$L_y$ values considered, the maximal point at which ${\cal B}$ crosses the real
$q$ axis, denoted $q_c$ is the same, and is equal to the value for the infinite
square lattice, $q_c=3$. This is the first family of strip graphs of which we
are aware that exhibits this type of universality of $q_c$.
