Abstract 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 $\begin{array}{} \Gamma_{G_n}^j \end{array}$ (z), for j = 1, 2, …, k; from these polynomials, we form a column vector Vn(z)=[ΓGn1(z),ΓGn2(z),…,ΓGnk(z)]t $\begin{array}{} V_n(z) = [\Gamma_{G_n}^1(z), \Gamma_{G_n}^2(z), \ldots, \Gamma_{G_n}^k(z)]^t \end{array}$ 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.

中文翻译：

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图的线性序列的属多项式的递归

摘要 给定一个有限图 H，H 线性序列的第 n 个成员 Gn 是通过添加边或识别顶点将 H 的不相交副本附加到 Gn-1 中 H 的最后一个副本来递归获得的，总是以相同的方式。图 G 的属多项式 ΓG(z) 是通过属枚举 G 的所有可定向嵌入的生成函数。在过去的 30 年中，大多数线性族图的多项式 ΓGn(z) 的计算都是通过将 Gn 的嵌入划分为类型 1、2、……、k 的多项式 ΓGnj $\begin{array}{ } \Gamma_{G_n}^j \end{array}$(z)，对于 j = 1, 2, ..., k; 从这些多项式，我们形成一个列向量 Vn(z)=[ΓGn1(z),ΓGn2(z),...,ΓGnk(z)]t $\begin{array}{} V_n(z) = [\Gamma_{ G_n}^1(z), \Gamma_{G_n}^2(z), \ldots, \Gamma_{G_n}^k(z)]^t \end{array}$ 满足递归 Vn(z) = M(z)Vn−1(z), 其中 M(z) 是 z 中多项式的 ak × k 矩阵。在本文中，Cayley-Hamilton 定理用于推导出 Γn(z) 的第 k 次线性递归，允许我们避免分区，从而将多项式的 k2 次乘法减少到 k 次这样的乘法。此外，该线性递归可以促进多项式的实根性和对数凹性的证明。我们用例子来说明。