A Tutte polynomial which distinguishes rooted unicyclic graphs

Let ΓρΓρ be a rooted graph. For any subset SS of E(Γ)E(Γ) let Cρ(S)Cρ(S) be the component of SS containing ρρ. Define r(S)=|V(Cρ(S))|−1r(S)=|V(Cρ(S))|−1, nulk(S)=|Cρ(S)|−r(S)nulk(S)=|Cρ(S)|−r(S), and nul(S)=|S|−r(S)nul(S)=|S|−r(S). With these, define the three-variable greedoid Tutte polynomial of ΓρΓρ, F(Γρ;t,p,q)F(Γρ;t,p,q) by: F(Γρ;t,p,q)=∑S⊆E(Γ)tr(E)−r(S)pnulk(S)qnul(S)−nulk(S).This polynomial generalizes the greedoid Tutte polynomial introduced in 1989 by Gordon and McMahon. Unlike the greedoid Tutte polynomial, the three-variable greedoid Tutte polynomial determines the number of gg-loops in the graph (loops and edges in a component of ΓΓ disjoint from the root). In addition, it is a complete invariant for the class of rooted loopless connected graphs which contain at most one cycle. As this is a polynomial of the greedoid underlying the rooted graph, we also generalize the polynomial to general greedoids.