An Algebraic Framework of Weighted Directed Graphs

We show that an algebraic formulation of weighted directed graphs leads to introducing a k-vector space equipped with two coproducts ∆ and˜∆ verifying the so-called coassociativity breaking equation (˜ ∆ ⊗ id)∆ = (id ⊗∆) ˜ ∆. Such a space is called an L-coalgebra. Explicit examples of such spaces are constructed and links between graph theory and coassociative coalgebras are given. 1. Introduction. On the one hand, motivated by periodicity phenomena in algebraic K-theory, Loday [12] introduced the notion of " noncommutative Lie algebra, " called Leibniz algebra. Such algebras D are described by a bracket [−,z] verifying the Leibniz identity: