Geodesic
Balanced Abelian group-valued functions on directed graphs
Ars Mathematica Contemporanea, Tome 13 (2017) no. 2, pp. 307-315.

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We discuss functions from the edges and vertices of a directed graph to an Abelian group. A function is called balanced if the sum of its values along any cycle is zero. The set of all balanced functions forms an Abelian group under addition. We study this group in two cases: when we are allowed to walk against the direction of an edge taking the opposite value of the function and when we are not allowed to walk against the direction.
DOI : 10.26493/1855-3974.910.460
Keywords: Consistent graphs, balanced signed graphs, balanced labelings of graphs, gain graphs, weighted graphs 
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Yonah Cherniavsky; Avraham Goldstein; Vadim E. Levit. Balanced Abelian group-valued functions on directed graphs. Ars Mathematica Contemporanea, Tome 13 (2017) no. 2, pp. 307-315. doi : 10.26493/1855-3974.910.460. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.910.460/

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