Computing the Tutte polynomial of Archimedean tilings

Abstract

We describe an algorithm to compute the Tutte polynomial of large fragments of Archimedean tilings by squares, triangles, hexagons and combinations thereof. Our algorithm improves a well known method for computing the Tutte polynomial of square lattices. We also address the problem of obtaining Tutte polynomial evaluations from the symbolic expressions generated by our algorithm, improving the best known lower bound for the asymptotics of the number of spanning forests, and the lower and upper bounds for the asymptotics of the number of acyclic orientations of the square lattice.

Asymptotic behavior of acyclic and cyclic orientations of directed lattice graphs

“…Denote N SF (G) as the number of spanning forests of a graph G and φ({G}) ≡ lim n(G)→∞ [N SF (G)] 1/n(G) . For example, for the square lattice, we have found 3.675183 ≤ φ(sq) < 3.699659, improving on the bounds 3.32 ≤ φ(sq) ≤ 3.8416195 in [20], the bounds 3.64497 ≤ φ(sq) ≤ 3.74101 in [21], the bounds 3.65166 ≤ φ(sq) ≤ 3.73635 in [26], and the upper bound φ(sq) ≤ 3.705603 in [25]. For the triangular and honeycomb lattices we obtain 5.393333 ≤ φ(tri) ≤ 5.494840 and 2.803787 ≤ φ(hc) ≤ 2.804781.…”

“…(In this context, it should be mentioned that our notation is different from the notation used in Refs. [20]- [26]; our quantities a(G), a 0 (G), and b(G) are the same as their α(G), α 0 (G), and β(G), respectively, and our quantities α({G}), α 0 ({G}), and β({G}) are the same as their quantities lim n(G)→∞ α(G) 1/n(G) , lim n(G)→∞ α 0 (G) 1/n(G) , and lim n(G)→∞ β(G) 1/n(G) , respectively. In [20], Merino and Welsh proved that Evidently, the new lower bound (3.6) that we have inferred is consistent with, and more restrictive than these previous lower bounds.…”

“…[7]- [19], some relevant previous studies of these quantities a(G), a 0 (G), and b(G) include Refs. [20]- [26]. In particular, in [22] we presented a number of results on acyclic orientations and their asymptotic behavior.…”