Exact chromatic polynomials for toroidal chains of complete graphs

We present exact calculations of the partition function of the zero-temperature Potts antiferromagnet (equivalently, the chromatic polynomial) for graphs of arbitrarily great length composed of repeated complete subgraphs K_b with b=5, 6 which have periodic or twisted periodic boundary condition in the longitudinal direction. In the L_x → ∞ limit, the continuous accumulation set of the chromatic zeros ℬ is determined. We give some results for arbitrary b including the extrema of the eigenvalues with coefficients of degree b-1 and the explicit forms of some classes of eigenvalues. We prove that the maximal point where ℬ crosses the real axis, q_c, satisfies the inequality q_c ≤ b for 2 ≤ b, the minimum value of q at which ℬ crosses the real q axis is q = 0, and we make a conjecture concerning the structure of the chromatic polynomial for Klein bottle strips.