We retain the indexing from Theorem 2.2.

Corollary 2.4 (Real and imaginary parts).Let n ≥ 3 and set m = n − 1 . For each nontrivial chromatic root z k = a k + i b k ,

where δ = 1 for even n and δ = 0 for odd n .

Proof. Writing z k − 1 = e i θ k with θ k = π ( 2 k + δ ) / m , Euler’s formula yields the result. □

Thus, the translated roots z k − 1 are equally spaced on the unit circle, with constant angular increment 2 π / m .

Corollary 2.5 (Real roots and symmetry).The nontrivial chromatic roots of C n occur in complex conjugate pairs. They include0for all n ≥ 3 , include2if and only if n is odd, and admit no other real values.

Proof. Real roots correspond to e i θ k ∈ { ± 1 } . The value −1 occurs for all m , while +1 occurs precisely when n is odd. □

Remark 2.6All nontrivial chromatic roots of C n lie onΓand hence in the rectangle 0 ≤ ℜ ( z ) ≤ 2 , | ℑ ( z ) | ≤ 1 , with extremal points0, 2, and 1 ± i .

This rigid geometry reflects the presence of a single dominant eigenvalue branch.

Beyond exact location, the roots of C n admit a simple asymptotic description: as n → ∞ they become uniformly distributed on Γ, and the associated logarithmic potential is elementary. Since these facts are classical and not used later, we record them briefly for context.

The nontrivial roots of P ( C n , z ) form an equally spaced set on Γ, up to a parity-dependent rotation. Consequently, the empirical root measures converge weakly to the uniform probability measure on Γ.

Theorem 2.7 (Equidistribution).The measures μ n = 1 n ∑ z ∈ ℛ n δ z converge weakly to the uniform measure σ onΓ.

Proof. This follows directly from the explicit parametrization z k = 1 + η ω n − 1 k and standard Riemann-sum convergence. □

Lemma 2.8 (Logarithmic potential).For all z ∈ ℂ ,

Proof. By rotational symmetry, the integral reduces to a classical mean-value computation on the unit circle. □

Theorem 2.9 (Potential convergence).For every z ∈ ℂ \ { 1 } ,

Proof. The closed form P ( C n , z ) = ( z − 1 ) n + ( − 1 ) n ( z − 1 ) yields the result by a direct comparison of exponential rates in the regimes | z − 1 | < 1 , | z − 1 | > 1 , and | z − 1 | = 1 . □

For cycles, the BKW limit set reduces to a single circle. In more complex recursive families, multiple eigenvalue branches compete, and the limiting geometry is dictated by nontrivial modulus ties. This transition motivates the prism analysis that follows.

Let G n = C n   □   P 2 denote the prism (cyclic ladder) graph, viewed as n rungs joining two horizontal n -cycles. A transfer matrix arises by propagating a proper coloring rung-by-rung while maintaining distinct colors on each rung.

Definition 3.1 (Rung state space).Fix q ≥ 0 and write [ q ] = { 1 , 2 , ⋯ , q } . Set

so that rung i is encoded by the ordered pair ( c ( u i ) , c ( v i ) ) ∈ Ω q .

The horizontal edges enforce c ( u i + 1 ) ≠ c ( u i ) and c ( v i + 1 ) ≠ c ( v i ) , while each rung enforces c ( u i + 1 ) ≠ c ( v i + 1 ) , i.e., membership in Ω q .

Definition 3.2 (Transfer matrix).For q ≥ 0 , define the matrix M ( q ) indexed by Ω q by

Figure 2. The prism graph G 8 = C 8   □   P 2 with 16 vertices and 24 edges. The graph consists of two concentric n -cycles (inner and outer) connected by n vertical edges (rungs). The Cartesian product structure C n   □   P 2 reflects the natural decomposition into horizontal cycles and vertical paths.

Proposition 3.3 (Trace formula).For every n ≥ 3 and every integer q ≥ 0 ,

In particular, for fixed n the right-hand side is a polynomial in q agreeing with P ( G n , q ) on q ∈ ℕ , hence the identity extends to all z ∈ ℂ .

Proof. A proper q -coloring of G n corresponds to a cyclic sequence of states

where each transition satisfies a i + 1 ≠ a i , b i + 1 ≠ b i , and a i + 1 ≠ b i + 1 . These are precisely the closed walks of length n in the directed transition graph on Ω q with adjacency matrix M ( q ) , counted by tr ( M ( q ) n ) . □

The eigen-branch expansion follows from the S q -symmetry given by relabeling colors.

Lemma 3.4 (Row-column subspace).Fix q ≥ 0 and write T q : = M ( q ) . Let W q be the subspace of functions f : Ω q → ℂ of the form f ( a , b ) = g ( a ) + h ( b ) with

Then W q is T q -invariant and, for every ( a , b ) ∈ Ω q ,

Equivalently, for each i ∈ [ q ] the pair ( g ( i ) , h ( i ) ) is updated by

whose eigenvalues are 1 − q and 3 − q , each yielding a ( q − 1 ) -dimensional mean-zero eigenspace in W q .

Proof. Fix ( a , b ) ∈ Ω q and write f ( a ′ , b ′ ) = g ( a ′ ) + h ( b ′ ) . Summing over admissible successors ( a ′ , b ′ ) with a ′ ≠ a , b ′ ≠ b , a ′ ≠ b ′ gives

For the g -sum, fix a ′ and count admissible b ′ . If a ′ = b then b ′ may be any element of [ q ] \ { b } , giving q − 1 choices; if a ′ ∉ { a , b } then b ′ must avoid { b , a ′ } , giving q − 2 choices. Hence

using ∑ g = 0 . By symmetry, the h -sum equals h ( a ) + ( 2 − q ) h ( b ) , yielding the stated formula.

Finally,

so the eigenvalues are 1 − q and 3 − q . Each eigendirection in ℂ 2 , together with the mean-zero condition, contributes a ( q − 1 ) -dimensional eigenspace in W q . □

We now identify the spectrum relevant to the trace.

Theorem 3.5 (Eigenvalue expansion).There exist four eigenvalue branches

with polynomial amplitudes

such that for all n ≥ 3 and all z ∈ ℂ ,

Proof. Work first at integer inputs z = q with q ≥ 4 , then extend by polynomial identity.

Fix q ≥ 4 and set V q = ℂ Ω q . Let T q be the operator with matrix M ( q ) , so tr ( M ( q ) n ) = tr ( T q n ) by Proposition 3.3. The symmetric group S q acts on Ω q by relabeling colors and commutes with T q , hence V q decomposes into S q -isotypic components on which T q acts scalarly.

On the fixed line ℂ 1 , each state ( a , b ) has ( q − 1 ) 2 − ( q − 2 ) = q 2 − 3 q + 3 admissible successors, so T q 1 = ( q 2 − 3 q + 3 ) 1 and λ 0 ( q ) = q 2 − 3 q + 3 .

The S q -stable subspace W q from Lemma 3.4 has dimension 2 ( q − 1 ) and splits into eigenspaces with eigenvalues λ 1 ( q ) = 1 − q and λ 2 ( q ) = 3 − q , each of multiplicity q − 1 .

Let U q be an S q -stable complement of span { 1 } ⊕ W q . Then dim ( U q ) = q ( q − 1 ) − 1 − 2 ( q − 1 ) = q 2 − 3 q + 1 . Choosing distinct a , b , c , d ∈ [ q ] and

one has ϕ ⊥ 1 and ϕ ⊥ W q , hence ϕ ∈ U q . A direct transition check gives T q ϕ = ϕ , so λ 3 ( q ) = 1 on U q .

Further, the ordered-pair permutation representation of S q on Ω q has exactly these four isotypic constituents [6]. Therefore

which matches (7) upon rewriting in terms of λ j ( q ) and α j ( q ) . For fixed n , both sides are polynomials in q that agree for all integers q ≥ 4 , hence are identical; evaluating at q = z gives (7) for all z ∈ ℂ . □

Remark 3.6 (Closed form and consistency checks).Substituting (5) and (6)into(7)yields

In particular, P ( G n , 0 ) = P ( G n , 1 ) = 0 for all n ≥ 3 . Small- n expansions provide checks only; the identity (7) is forced by the trace formula together with the S q -reduction producing the four eigen-branches.

The representation (7) is a finite exponential sum in n . Consequently, root accumulation can occur only where at least two branches tie in modulus at the dominant scale, or at isolated points where a uniquely dominant branch has vanishing amplitude. The Beraha-Kahane-Weiss theorem makes this localization precise.

Theorem 3.7 (BKW accumulation set for prism chromatic roots).Let ℒ be the set of accumulation points in ℂ of chromatic roots of the prism family { G n } n ≥ 3 . With

as in Theorem3.5, define

Then:

1) ℒ ⊆ ℰ ∪ A .

2) If z 0 ∈ ℰ satisfies

α i ( z 0 ) α j ( z 0 ) ≠ 0 , and λ i / λ j is not locally constant near z 0 , then z 0 ∈ ℒ .

For the prism branches (5), the elementary tie loci are

The amplitude zeros are

Thus, on any subset of a tie locus where exactly two branches dominate and the tied amplitudes are nonzero, the points belong to ℒ by (ii).

Proof. (i) Fix z 0 ∈ ℂ and assume a unique index j 0 satisfies

By continuity, there exists a neighborhood U of z 0 and constants c > 0 and ρ ∈ ( 0 , 1 ) such that | α j 0 ( z ) | ≥ c and | λ i ( z ) | ≤ ρ | λ j 0 ( z ) | for all z ∈ U and i ≠ j 0 . Factoring the dominant branch gives

On U one has | R n ( z ) | ≤ C ρ n uniformly for some C > 0 , hence 1 + R n ( z ) ≠ 0 on U for all sufficiently large n . Thus z 0 cannot be an accumulation point of zeros, proving ℒ ⊆ ℰ ∪ A .

(ii) Let z 0 ∈ ℰ with exactly two dominant branches i ≠ j and α i ( z 0 ) α j ( z 0 ) ≠ 0 . Shrinking to a neighborhood where all remaining branches are uniformly subdominant, write

uniformly for some ρ ∈ ( 0 , 1 ) . Since λ i / λ j is analytic and not locally constant, its argument varies in every neighborhood of z 0 , and the two leading terms attain near-opposition for infinitely many n . The standard BKW argument for exponential sums then yields zeros of P ( G n , ⋅ ) arbitrarily close to z 0 for arbitrarily large n , hence z 0 ∈ ℒ .

Finally, (11)-(14) and (15) follow by direct substitution from (5) and (6). □

We now specialize the Beraha-Kahane-Weiss description to the explicit prism eigen-branches, isolating the subsets that can contribute to the limiting chromatic root set.

Corollary 3.8 (Prism limiting set).Let λ 0 , λ 1 , λ 2 , λ 3 and α 0 , α 1 , α 2 , α 3 be as in Theorem 3.5. For 0 ≤ i < j ≤ 3 define

and define

Then

Moreover, if z 0 ∈ ℰ i j is a point at which exactly the two branches i , j are dominant and λ i / λ j is not locally constant near z 0 , then z 0 ∈ ℒ .

This is an immediate specialization of Theorem 3.7 to the eigenvalues (5) and amplitudes (6), with the dominance conditions made explicit.

For G n = C n   □   P 2 the eigen-branch expansion in Theorem 3.5 reads

The ties among λ 1 , λ 2 , λ 3 are elementary:

and together with C 03 = { | λ 0 ( z ) | = 1 } they account for all ties involving λ 3 . The remaining balances tie λ 0 with λ 1 or λ 2 and yield quartic real-algebraic curves.

Theorem 3.9(Quartic modulus ties for λ 0 ).Let z = x + i y with x , y ∈ ℝ , and set

The modulus-tie sets

are quartic real-algebraic curves given by

Proof. Writing z = x + i y gives

so

Since | λ 1 ( z ) | 2 = ( x − 1 ) 2 + y 2 and | λ 2 ( z ) | 2 = ( x − 3 ) 2 + y 2 , the equalities | λ 0 | = | λ 1 | and | λ 0 | = | λ 2 | are equivalent to (16) and (17). Each is polynomial in ( x , y ) and has total degree four. □

Corollary 3.10 (Basic properties).The quartic curves C 01 and C 02 satisfy:

1) C 01 ∩ ℝ = { 2 } , with even intersection multiplicity;

2) C 02 ∩ ℝ = { 0 , 2 } ;

3) both curves are invariant under complex conjugation.

Proof. Setting y = 0 in (16) gives ( x 2 − 3 x + 3 ) 2 = ( x − 1 ) 2 , hence x 2 − 3 x + 3 = ± ( x − 1 ) . The + sign yields ( x − 2 ) 2 = 0 ; the − sign yields x 2 − 2 x + 2 = 0 with no real solutions. Thus C 01 ∩ ℝ = { 2 } with even multiplicity.

Setting y = 0 in (17) gives ( x 2 − 3 x + 3 ) 2 = ( x − 3 ) 2 , hence x 2 − 3 x + 3 = ± ( x − 3 ) . The − sign yields x ( x − 2 ) = 0 , giving { 0 , 2 } ; the + sign has no real solutions. Both equations depend on y only via y 2 , so they are invariant under conjugation. □

Remark 3.11 (Dominance on quartic ties).On C 01 we have | λ 0 | = | λ 1 | = | z − 1 | . The linear comparison

implies that for every z ∈ C 01 with ℜ ( z ) ≤ 1 , | z − 3 | ≥ | z − 1 | = | λ 0 | , so λ 0 cannot be strictly dominant there. On C 02 we have | λ 0 | = | λ 2 | = | z − 3 | , and

implies that for every z ∈ C 02 with ℜ ( z ) ≥ 2 , | z − 1 | ≥ | z − 3 | = | λ 0 | , again excluding strict dominance of λ 0 on that portion. In these regions the quartic ties cannot contribute to accumulation arcs.

Equations (16) (17) also sharpen the tie geometry beyond what contour plots alone resolve: tangencies and near-intersections can be checked algebraically, and the dominance inequalities can be verified directly without relying on numerical resolution.