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Motivated by Stanley's $\mathbf{(3+1)}$-free conjecture on chromaticsymmetric functions, Foley, Ho\`{a}ng and Merkel introduced the concept ofstrong $e$-positivity and conjectured that a graph is strongly $e$-positive ifand only if it is (claw, net)-free. In order to study strongly $e$-positivegraphs, they further introduced the twinning operation on a graph $G$ withrespect 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 ofthem. Foley, Ho\`{a}ng and Merkel conjectured that if $G$ is $e$-positive, thenso is the resulting twin graph $G_v$ for any vertex $v$. Based on the theory ofchromatic symmetric functions in non-commuting variables developed by Gebhardand Sagan, we establish the $e$-positivity of a class of graphs called tadpolegraphs. By considering the twinning operation on a subclass of these graphswith respect to certain vertices we disprove the latter conjecture of Foley,Ho\`{a}ng and Merkel. We further show that if $G$ is $e$-positive, the twingraph $G_v$ and more generally the clan graphs $G^{(k)}_v$ ($k \ge 1$) may noteven be $s$-positive, where $G^{(k)}_v$ is obtained from $G$ by applying $k$twinning operations to $v$.

The twinning operation on graphs does not always preserve $e$-positivity