Csikv\'ari's poset and Tutte polynomial
Abstract
Csikv\'ari constructed a poset on trees to prove that several graph functions attain extreme values at the star and the path among the trees on a fixed number of vertices. Reiner and Smith proved that the Tutte polynomials T(1,y) of cones over trees, which are the graphs obtained by attaching a cone vertex to a tree, have the described extreme behavior. They further conjectured that the result can be strengthened in terms of Csikv\'ari's poset. We solve this conjecture affirmatively.
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