Explicit Equimodular Curves for Prism-Graph Chromatic Polynomials and the Beraha-Kahane-Weiss Limit Set

Abstract

For the prism (cyclic ladder) graphs G n = C n □ P 2 , the chromatic polynomial admits a four-branch transfer-matrix expansion P ( G n , z ) = ∑ j = 0 3 α j ( z ) λ j ( z ) n , λ 0 ( z ) = z 2 − 3 z + 3 , λ 1 ( z ) = 1 − z , λ 2 ( z ) = 3 − z , λ 3 ( z ) = 1 , with explicit polynomial amplitudes α j ( z ) . By the Beraha-Kahane-Weiss mechanism, accumulation of chromatic roots as n → ∞ is confined to loci where two or more dominant eigenvalues tie in modulus, together with isolated points arising from vanishing dominant amplitudes. We give a complete real-algebraic description of these modulus-tie sets for the prism family, including closed-form Cartesian quartic equations for the quadratic-linear balances | z 2 − 3 z + 3 | = | z − 1 | and | z 2 − 3 z + 3 | = | z − 3 | . These identities replace plot-based equimodular boundaries with verifiable equations and allow direct symbolic certification of the dominance inequalities governing the BKW accumulation arcs. For comparison, we also recall the cycle family C n , whose nontrivial chromatic roots lie on the circle | z − 1 | = 1 and are uniformly distributed in angle.