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

Motivated by Stanley and Stembridge's (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 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. By considering the twinning operation on a subclass of tadpole graphs with respect to certain vertices we disprove the latter conjecture. We further show that if G is e-positive, the twin graph G_v and more generally the clan graphs G_v ^(k) (k ≥ 1) may not even be s-positive, where G_v ^(k) is obtained from G by applying k twinning operations to v.