论文标题
集团间倒数对
Group-Graph Reciprocal Pairs
论文作者
论文摘要
在2018年的论文中，卡梅伦和Semeraro提出了找到所有集体图互倒数对的问题。在本文中，我们为寻找所有这些对成对做出了重要贡献。 A group and graph form a reciprocal pair if they satisfy the relation $$P_{Γ,G}(x)=(-1)^nF_G(-x)$$ where $P_{Γ,G}(x)$ is the orbital chromatic polynomial of a graph $Γ$ and $F_G(x)$ is the cycle polynomial of a finite permutation group.我们将一组图表定义为\ textit {$ k $ -stars}，并证明它们根据$ k $的不同组满足了与某些组的相互关系。这些图由一个完整的图形组成，带有$ K $顶点和另外一个$α$“点”，该图仅连接到中心的每个顶点。该组是$ s_k \ timess_α$的子组，这是\ textit {$ k $ -star}的自动形态组，而$α$是恒星中的点数。我们猜想了群体互倒数对的列表。
In a 2018 paper, Cameron and Semeraro posed the problem of finding all group-graph reciprocal pairs. In this paper, we make a significant contribution to finding all such pairs. A group and graph form a reciprocal pair if they satisfy the relation $$P_{Γ,G}(x)=(-1)^nF_G(-x)$$ where $P_{Γ,G}(x)$ is the orbital chromatic polynomial of a graph $Γ$ and $F_G(x)$ is the cycle polynomial of a finite permutation group. We define a set of graphs to be \textit{$k$-stars} and prove that they satisfy a reciprocality relation with some group depending on $k$. These graphs are comprised of a complete graph with $k$ vertices and a further $α$ `points' which are only connected to each vertex in the centre. This group is a subgroup of $S_k\times S_α$, which is the automorphism group of a \textit{$k$-star} and $α$ is the number of points on the star. We conjecture a list of group-graph reciprocal pairs.
