Recurrences for the genus polynomials of linear sequences of graphs

Given a finite graph H, the n^th member G_n of an H-linear sequence is obtained recursively by attaching a disjoint copy of H to the last copy of H in G_n-1 by adding edges or identifying vertices, always in the same way. The genus polynomial Γ_G(z) of a graph G is the generating function enumerating all orientable embeddings of G by genus. Over the past 30 years, most calculations of genus polynomials Γ_Gn(z) for the graphs in a linear family have been obtained by partitioning the embeddings of G_n into types 1, 2, ⋯, k with polynomials ΓG_n^j (z), for j = 1, 2, ⋯, k; from these polynomials, we form a column vector V_n(z)=[ΓG_n¹(z),ΓG_n²(z), that satisfies a recursion V_n(z) = M(z)V_n-1(z), where M(z) is a k × k matrix of polynomials in z. In this paper, the Cayley-Hamilton theorem is used to derive a k^th degree linear recursion for Γ_n(z), allowing us to avoid the partitioning, thereby yielding a reduction from k² multiplications of polynomials to k such multiplications. Moreover, that linear recursion can facilitate proofs of real-rootedness and log-concavity of the polynomials. We illustrate with examples.