论文标题
图上的孪晶操作并不总是保留$ e $ - 积极性
The twinning operation on graphs does not always preserve $e$-positivity
论文作者
论文摘要
由斯坦利的$ \ mathbf {（3+1）} $ - 在色度对称功能上的自由猜想，Foley，Hoàng和Merkel引入了强烈$ e $ positivity的概念，并猜测图认为图形是强烈的$ e $ $ e $ positive，并且仅在IT（Claw，网络，网上）时才呈现。为了强烈研究$ e $ p的阳性图，他们进一步在图形$ g $上引入了twinning操作，相对于顶点$ v $，它增加了$ v $ g $ g $ g $ v $ v $ v $ v $和$ v'$，因此$ v $和$ v'$均与其他任何顶点相邻。 Foley，Hoàng和Merkel猜想，如果$ G $是$ e $ - 阳性，那么任何顶点$ v $的生成的双图$ g_v $也是如此。基于Gebhard和Sagan开发的非交通变量中的色度对称函数的理论，我们建立了称为Tadpole图的一类图形的$ e $ potititive。通过考虑这些图的子类的孪生操作，相对于某些顶点，我们反驳了Foley，Hoàng和Merkel的后一种猜想。我们进一步表明，如果$ g $是$ e $ - 良好的，则双图$ g_v $，更一般而言，氏族图$ g^{（k）} _v $（$ k \ ge 1 $）甚至都不是$ s $ positive，其中$ g^{（k）v $从$ g $中获得$ g $ a $ k $ a $ k $ a $ k $ $ k $ $ k $ $ k $ $ k $ $ k $ twinning twning to $ $ twning to vinning
Motivated by Stanley's $\mathbf{(3+1)}$-free conjecture on chromatic symmetric functions, Foley, Hoàng and Merkel introduced the concept of strong $e$-positivity and conjectured that a graph is strongly $e$-positive if and only if it is (claw, net)-free. In order to study strongly $e$-positive graphs, they further introduced the twinning operation on a graph $G$ with respect to a vertex $v$, which adds a vertex $v'$ to $G$ such that $v$ and $v'$ are adjacent and any other vertex is adjacent to both of them or neither of them. Foley, Hoàng and Merkel conjectured that if $G$ is $e$-positive, then so is the resulting twin graph $G_v$ for any vertex $v$. Based on the theory of chromatic symmetric functions in non-commuting variables developed by Gebhard and Sagan, we establish the $e$-positivity of a class of graphs called tadpole graphs. By considering the twinning operation on a subclass of these graphs with respect to certain vertices we disprove the latter conjecture of Foley, Hoàng and Merkel. We further show that if $G$ is $e$-positive, the twin graph $G_v$ and more generally the clan graphs $G^{(k)}_v$ ($k \ge 1$) may not even be $s$-positive, where $G^{(k)}_v$ is obtained from $G$ by applying $k$ twinning operations to $v$.
