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.

The bichromatic triangle polynomial $P_G\left(k\right)$ counts vertex k-colorings in which every triangle uses exactly two colors. We develop a transfer matrix framework for three canonical families of 2-trees: book graphs $B_n$, 1-fans $F_n^1$, and triangulated ladders $T{L}_m$. In each case, $P_G\left(k\right)$ satisfies a second-order linear recurrence with an explicit closed form; for $T{L}_m$ this yields a Chebyshev representation, while for $F_n^1$ the binary specialization gives $P_{F_n^1}\left(2\right)=2F_{n+1}$. A spectral identity ${\alpha }^2=r_{+}$ links the dominant characteristic roots of the fan and ladder recurrences, implying identical exponential growth rates when indexed by vertex count, whereas book graphs grow strictly faster for $k\ge 4$. In fact, this correspondence is exact: for all $k\ge 2$, the triangulated ladder polynomial coincides with that of a suitably indexed 1-fan. Passing to line graphs, we interpret $P_{L\left(K_n\right)}\left(k\right)$ as counting edge colorings of $K_n$ that forbid both monochromatic and rainbow triangles, and we identify a sharp obstruction threshold at $n\ge 6$.

We establish explicit formulas for Bell numbers and graphical Stirling numbers of complete multipartite graphs, complete bipartite graphs with removed perfect matchings, and Mycielskian trees. For complete multipartite graphs $K(n_1,\ldots,n_\ell)$, we provide a simplified proof that $B(G) = \prod_{i=1}^\ell \bell{n_i}$. We derive $B(K_{n,n} - M) = \sum_{k=0}^{n} \binom{n}{k} \bell{k}^2$ for removed perfect matching $M$, and for Mycielskian star graphs, $B(M(St_n); 3) = 2^n + 1$ and $B(M(St_n); 2n) = 2n^2 - 3n + 3$. Results extend to Mycielskians of arbitrary trees. Our computational verifications establish links between graphical Bell numbers and fundamental sequences in combinatorics and pattern avoidance, including identification of several OEIS entries: A000051, A096376, A116735, A384980, A384981, A384988, A385432, and A385437.

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.

Explicit formulas for graphical Stirling and Bell numbers are known for relatively few graph families. We derive exact expressions for three classes whose independence structure admits a complete combinatorial description: complete multipartite graphs, the graph obtained from a balanced complete bipartite graph by deleting a perfect matching, and the Mycielskian of a star. For complete multipartite graphs we express the graphical Stirling number as a convolution of classical Stirling numbers across the partite classes, and we recover the known factorization of the graphical Bell number as a product of classical Bell numbers. For the matching-deleted graph we show that its graphical Bell number is a binomial convolution of squared Bell numbers, which we identify as a moment of a product of two independent Poisson random variables with unit mean. This representation yields log-convexity of the sequence, a sharp exponential lower bound, a two-sided estimate, and a Laplace-transform identity. For the Mycielskian of a star, a decomposition according to the block containing the original center vertex, together with Vandermonde’s convolution and a Stirling recurrence, gives a single-sum closed form for the graphical Stirling numbers, from which two explicit evaluations follow. Several resulting integer sequences appear in the OEIS, and one Bell-number sequence appears not to be currently recorded there.