A Tutte polynomial which distinguishes rooted unicyclic graphs

Let Γ ρ be a rooted graph. For any subset S of E ( Γ ) let C ρ ( S ) be the component of S containing ρ . Define r ( S ) = | V ( C ρ ( S ) ) | − 1 , nul k ( S ) = | C ρ ( S ) | − r ( S ) , and nul ( S ) = | S | − r ( S ) . With these, define the three-variable greedoid Tutte polynomial of Γ ρ , F ( Γ ρ ; t , p , q ) by: F ( Γ ρ ; t , p , q ) = ∑ S ⊆ E ( Γ ) t r ( E ) − r ( S ) p nul k ( S ) q nul ( S ) − nul k ( S ) . olynomial 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 g -loops in the graph (loops and edges in a component of Γ 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.