International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 58, Pages 3657-3678

An algebraic framework of weighted directed graphs

Philippe Leroux

Institut de Recherche en Mathématiques de Rennes, Université de Rennes 1 et UMR 6625 CNRS, Campus de Beaulieu, Rennes Cedex 35042, France

Received 19 February 2003

Copyright © 2003 Philippe Leroux. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We show that an algebraic formulation of weighted directed graphs leads to introducing a k-vector space equipped with two coproducts Δ and Δ˜ verifying the so-called coassociativity breaking equation (Δ˜id)Δ=(idΔ)Δ˜. 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.