The Twinning Operation on Graphs Does not Always Preserve $e$-positivity

Motivated by Stanley and Stembridge's $(\mathbf{3}+\mathbf{1})$-free conjecture on chromatic symmetric functions, Foley, Hoàng Merkel introduced the concept of strong $e$-positivity conjectured that a graph is strongly $e$-positive if only it (claw, net)-free. In order to study graphs, they twinning operation $G$ with respect vertex $v$, which adds $v'$ such $v$ are adjacent any other both them or neither them. $e$-positive, then so resulting twin $G_v$ for $v$. By considering subclass tadpole graphs certain vertices we disprove latter conjecture. We further show more generally clan $G^{(k)}_v$ ($k \geq 1$) may not even be $s$-positive, where obtained from applying $k$ operations