Talk:Gain graph
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[edit] Group action?
Does a gain graph satisfy at least some of the axioms of a group action? So, for example, if g,h,k are group elements, with k=gh, and if v a vertex, then, if the edge kv exists, and the edge hv exists and if the edge g(hv) exists, then will kv be the same vertex as g(hv) ? If so, then it seems to me that a gain graph is like a group action
except that it is only a partial function on the set of pairs 
. Is that correct?
If this holds, it seems like one could legitimately call a grain graph a "partial group action", right?
If this does not hold, then it is not at all clear (to me) why one is justified in calling the edge labels "group elements"; one could just as easily call then "foobar elements", and that would be a meaningless label unless the axioms of a foobar were to be made to hold ... linas (talk) 21:23, 24 March 2008 (UTC)
- The group elements don't label vertices, they label edges. Thus, there's no such edge as kv. An edge has two end vertices, so you might describe an edge as uv, where u and v are vertices. This edge will have a gain (group element), say g. However, uv isn't the best notation for an edge because there could be many edges uv, each with a different gain. I like the notation e:uv which distinguishes e from another edge f:uv with the same endpoints.
- The multiplication takes place between consecutive edges in a path (or walk) in the graph. Suppose you have an edge uv with gain g, and an edge vw with gain h. Then the path uvw composed of those two edges has gain gh=k. I hope this answers your question.
