TITLE:
Explicit Equimodular Curves for Prism-Graph Chromatic Polynomials and the Beraha-Kahane-Weiss Limit Set
AUTHORS:
Julian Allagan, Vitaly Voloshin, Vladimir Deriglazov, Rogelio N. Lopez-Bonilla
KEYWORDS:
Chromatic Polynomial, Chromatic Roots, Prism Graphs, Cyclic Ladders, Beraha-Kahane-Weiss Theorem, Equimodular Curves, Transfer Matrix, Graph Families, Real-Algebraic Curves
JOURNAL NAME:
American Journal of Computational Mathematics,
Vol.16 No.1,
March
25,
2026
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
−3z+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
−3z+3 |=|
z−1 |
and
|
z
2
−3z+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.