AI 中文总结
研究三次图的亏格多项式，通过展示共享多种参数但亏格多项式不同的三次图，解释差异原因，构造共谱三次对，计算特定顶点数内连通三次图的亏格多项式并推导最小亏格下界。
AI 中文摘要
图的可定向亏格多项式通过亏格来计算其胞腔嵌入。对于有限简单2 - 连通三次图，它是一个圈拟阵不变量：\(M(G)\cong M(H)\)意味着\(\Gamma_G=\Gamma_H\)。邻接谱和亏格多项式不可比。我们展示了16个顶点的连通三次图，它们共享邻接谱、生成树数量、围长、直径、顶点和边连通性、自同构群阶以及长度为10的圈数，但亏格多项式两两不同。将预期面数在围长两倍处拆分可解释差异：短面是谱的，长面不是。我们构造了一个明确的无限族连通共谱三次对\((G_t,H_t)\)，其顶点数为\(14 + 2t\)且最小亏格不同。我们还计算了所有22个顶点以内的7,875,918个连通三次图的亏格多项式，并从短圈数推导出最小亏格的确定性下界。
英文摘要
The orientable genus polynomial of a graph counts its cellular embeddings by genus. For finite simple $2$-connected cubic graphs it is a cycle-matroid invariant: $M(G)\cong M(H)$ implies $Γ_G=Γ_H$. The adjacency spectrum and the genus polynomial are incomparable: neither determines the other. We exhibit connected cubic graphs on $16$ vertices sharing the adjacency spectrum, spanning-tree count, girth, diameter, vertex and edge connectivity, automorphism-group order, and cycle counts through length $10$, yet with pairwise distinct genus polynomials. Splitting the expected face count at twice the girth explains the difference: short faces are spectral, long faces are not. We construct an explicit infinite family of connected cospectral cubic pairs $(G_t,H_t)$ on $14+2t$ vertices whose minimum genera differ. We also compute the genus polynomials of all $7,875,918$ connected cubic graphs through $22$ vertices and derive from short-cycle counts a deterministic lower bound on the minimum genus.
