We study limiting chromatic-root loci for several recursively constructed prism-derived graph families by means of finite transfer-matrix reductions. For each family, the chromatic polynomial admits a spectral expansion $P(G_n,z)={\sum }_j{\alpha }_j\left(z\right){\lambda }_j{\left(z\right)}^n$, so the Beraha–Kahane–Weiss theorem reduces the asymptotic analysis to dominant spectral competition and residual amplitude cancellation. For the antiprism family $A_n$, the reduced transfer operator has discriminant $\Delta \left(z\right)=9-4z$. The unit-modulus conditions for the two nontrivial spectral branches pull back to the irreducible quartic $x^4-8x^3+2x^2{y}^2+21x^2-8x{y}^2-18x+y^4+5y^2=0$. We prove that this entire quartic is the equimodular accumulation locus for $A_n$, with the node at $z=3$ separating the two dominance types. The same quartic arises for the circulant family $C_n\left(1,2\right)$ through the same reduced spectral normal form. We also prove that $A_n$ has a nonfixed isolated accumulation point at the fifth Beraha number $B_5=(3+\sqrt{5})/2$, and that $C_n\left(1,2\right)$ has the same isolated point along its even subsequence after the residual branches $+1$ and $-1$ are combined by parity. In contrast, for $C_n\left(1,3\right)$, the identity ${\lambda }_{+}\left(z\right){\lambda }_{-}\left(z\right)=-1$ collapses the reduced-sector equimodular locus to the vertical segment $\mathfrak{R}\left(z\right)=2$, $|\mathfrak{I}(z)|\le 2$. For the generalized Petersen family $G(n,2)$, we obtain an irreducible cubic equation for one invariant sector, indicating the first obstruction to the quadratic pullback method while leaving the full branch expansion open.

We develop a transfer-matrix framework for the asymptotic analysis of chromatic polynomials of recursively constructed prism-derived graph families. For each family {G_n}, the chromatic polynomial admits a finite spectral expansion P(G_n, z) = ∑_j α_j(z)λ_j(z)ⁿ, and the Beraha–Kahane–Weiss theorem identifies the limiting chromatic-root set through competition among dominant spectral branches. For the antiprism family A_n, the reduced transfer operator has discriminant ∆(z) = 9 − 4z. After the substitution w² = 9 − 4z, the unit-modulus conditions for the spectral branches λ_± pull back to the irreducible quartic curve x⁴ − 8x³ + 2x²y² + 21x² − 8xy² − 18x + y⁴ + 5y² = 0. The same quartic arises independently for the circulant family C_n({1, 2}) through an identical reduced spectral normal form. In contrast, the chord family C_n({1, 3}) satisfies λ+(z)λ−(z) = −1, forcing collapse of the reduced equimodular locus to the vertical segment ℜ(z) = 2, |ℑ(z)| ≤ 2. We further prove that the antiprism family possesses a nonfixed isolated accumulation point at the fifth Beraha number B₅ = (3 + √5)/2, arising from residual amplitude cancellation under unique dominant spectral behavior. The generalized Petersen family G(n, 2) exhibits an irreducible cubic spectral equation, demonstrating an obstruction to the quadratic pullback mechanism. These examples reveal a structural relationship between reduced transfer-matrix spectra and the algebraic geometry of limiting chromatic- root loci, showing that distinct recursive graph families may share identical asymptotic geometry through common reduced spectral normal forms.

We study generalized circular chord graphs $\mathcal C^{(k)}_n$, formed from a cycle $C_n$ by adding fixed-offset chords of length $k$ and, for even $n$, diameters. Using transfer matrix methods, we derive exact formulas for 3-colorings when $k=3$: for odd $n$, we obtain \[ P(\mathcal{C}_n^{(3)},3) = L_n + 2\cos\left(\frac{2πn}{3}\right) + 2s_n + 2 \] where $L_n$ is the Lucas sequence and $(s_n)$ satisfies $s_{n+3} = -s_{n+2} - s_n$, yielding golden-ratio asymptotic growth $\varphi^n + O(ρ^n)$ along odd indices. For even $n$, we construct a paired-window transfer matrix that exactly enumerates $P(\mathcal{C}_{2m}^{(3)},3)$ while capturing diameter constraints. The chromatic counts exhibit pronounced modular patterns across residue classes without universal vanishing rules (see OEIS A383733). We provide efficient algorithms for exact enumeration and demonstrate applications to cyclic scheduling problems where these results serve as feasibility engines for airline gate assignment, wireless sensor networks, and multiprocessor task coordination.

The chromatic polynomial of a 2-tree depends only on its order and therefore does not distinguish many non-isomorphic canonical families. We introduce a multivariate edge-weighted bichromatic partition function $P_G(q,v,w)$ in which every triangular face is constrained to use exactly two colors, while v and w record monochromatic and bichromatic edge contributions. Our main result concerns canonical recursive 2-tree families generated by a single active-edge interface. For this class of constructions, the extension behavior depends only on whether the interface vertices received equal or different colors, leading to transfer matrices of a fixed $2\times 2$ size independent of q. This leads to a transfer-word representation for active-edge constructions and explicit partition functions for several canonical 2-tree families. As consequences, we obtain the partition-function identity $P_{S_n}(q,v,w)=P_{F_n^1}(q,v,w)$, equating two families that are non-isomorphic for $n\ge 6$, together with the operator factorization $M_{\mathrm{ladder}}=M_{\mathrm{snake}}^2$, from which $P_{T{L}_m}(q,v,w)=P_{S_{2m+2}}(q,v,w)$ follows; the latter equality is induced by the graph isomorphism $T{L}_m\cong S_{2m+2}$. For $q\ge 2$ and $v,w>0$, we determine the associated spectral growth rates and show that the Book and Snake transfer systems have equal spectral radii precisely on the locus $(q-1)w=2v$. Under the same positivity assumptions, this is also the unique parameter locus on which the elementary transfer operators commute.