The twinning operation on graphs does not always preserve \(e\)-positivity

Motivated by Stanley's \(\mathbf{(3+1)}\)-free conjecture on chromatic symmetric functions, Foley, Ho\`{a}ng 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, Ho\`{a}ng 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, Ho\`{a}ng 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 \(v\).