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Is there a mathematical, graph theory, game theory term for a graph that is fully connected and balanced evenly with each other node.

I'm thinking in situations like Rock paper scissors where each action is equally balanced by each other option?

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What you ask is not very clear. Perhaps this will help,though: mathworld.wolfram.com/Vertex-TransitiveGraph.html –  Aryabhata Jul 12 '11 at 21:33
    
A directed graph which is connected, and where every vertex has the same indegree and outdegree, is called Eulerian mathworld.wolfram.com/EulerianGraph.html . Is that what you're looking for? –  David Speyer Jul 13 '11 at 2:28
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A regular tournament $T=(V,E)$ is a complete digraph on $|V|=2n+1$ vertices where each vertex has the same score as each other vertex. Each vertex is connected to each other vertex by exactly one directed edge and each vertex dominates exactly $\frac{|V|-1}{2}$ other edges and is dominated by the $\frac{|V|-1}{2}$ other edges.

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