Balanced vertex weightings and arborescences

Abstract Let D be a digraph with vertex set V and let (Γ, +, 0) be any Abelian group. A weighting b of the vertices with elements of Γ is balanced if for every vertex v the weight on v is “the average” of the weights on the vertices in the out-neighborhood of v , i.e., d + ( v ) b ( v ) = Σ b ( w ) where the summation is over all vertices w in the out-neighborhood of v and d + ( v ) denotes the out-degree of v . The set B ( Γ ) of all balanced vertex weightings determines a group. Let a v denote the number of spanning in arborescences rooted at vertex v and let α = gcd {a v ¦v ∈ V} . In this paper we show that α has a unique factorization α = α 1 α 2 … α m such that α, is a multiple of α i +1 , i = 1, 2,…, m − 1 and such that for every Abelian group Γ B ( Γ ) ≅ Γ × Γ ( a 1 ) × … × Γ ( α m ), where Γ(α i ) = {g ∈ Γ ¦ α i g = 0} . As a corollary, we obtain the characterization of the group of bicycles over Γ given in (Berman, SIAM J. Algebraic Discrete Methods 7 (1986), 1–12). We also obtain many results about the arborescence numbers including a new proof of Tutte's trinity theorem and two formulae for a D .