论文标题
具有非现实根的立方图的多项式属
Genus Polynomials of Cubic Graphs with Non-Real Roots
论文作者
论文摘要
给定图$ g $，其多项式属为$γ_g（x）= \ sum_ {k \ geq 0} g_k（g）x^k $，其中$ g_k（g）$是在可定向表面的$ g $ g $ k $ k $的2细胞嵌入$ g $的2个细胞嵌入数量。对数coveity属分布（LCGD）的猜想指出，每个图的多项式属都是对数 - concave。 Stahl进一步猜想，每个图的多项式属仅具有真实的根，但是后来被反驳了。我们确定了几种立方图的示例，除了至少一个非现实的根外，多项式属的属具有一个二次因子，该因子是非conconcave的，当纳入实际数字上时。
Given a graph $G$, its genus polynomial is $Γ_G(x) = \sum_{k\geq 0} g_k(G)x^k$, where $g_k(G)$ is the number of 2-cell embeddings of $G$ in an orientable surface of genus $k$. The Log-Concavity Genus Distribution (LCGD) Conjecture states that the genus polynomial of every graph is log-concave. It was further conjectured by Stahl that the genus polynomial of every graph has only real roots, however this was later disproved. We identify several examples of cubic graphs whose genus polynomials, in addition to having at least one non-real root, have a quadratic factor that is non-log-concave when factored over the real numbers.
