Given a finite graph H, the nth member Gn of an H-linear sequence is obtained recursively by attaching a disjoint copy of H to the last copy of H in Gn-1 by adding edges or identifying vertices, always in the same way. The genus polynomial ΓG(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 ΓGn(z) for the graphs in a linear family have been obtained by partitioning the embeddings of Gn into types 1, 2, ⋯, k with polynomials ΓGnj (z), for j = 1, 2, ⋯, k; from these polynomials, we form a column vector Vn(z)=[ΓGn1(z),ΓGn2(z), that satisfies a recursion Vn(z) = M(z)Vn-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 kth degree linear recursion for Γn(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.
- genus polynomial
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