On the Limiting Chromatic Zero Set of Triangulated Prism Graphs and Boundary Invariance

In this paper, we study the asymptotic distribution of the zeros of the chromatic polynomial P(G_n, q) in the complex plane C for a one-parameter family of triangulated prism graphs, Y_n*, formed by adding a set of diagonal edges to the Cartesian product C_n □ K_2. Using the transfer-matrix method, we derive a closed-form representation for P(Y_n*, q) in terms of the eigenvalues of a reduced 2 × 2 transfer matrix M(q). Applying the Beraha-Kahane-Weiss (BKW) theorem, we rigorously characterize the limiting locus B(Y*) of complex chromatic zeros as n → ∞ via equimodular curves |λ_1(q)| = |λ_2(q)|. Furthermore, we evaluate the chromatic critical points along the real axis R. Finally, we prove that while Möbius boundary conditions (antiperiodic boundary conditions) alter the finite n-zero locations, the limiting continuous locus B(Y*) remains strictly invariant under Möbius twists, demonstrating that the limiting zero set is governed entirely by the local structure of the transfer operator.