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Are there any practical situations that can be represented using a modified definition of an edge - where each edge connects three vertices instead of two?

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    Sure, for example the notion of three points being collinear. See http://en.wikipedia.org/wiki/Hypergraph .2012-08-12
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    Would a triangle do? It sort of connects 3 vertices. Another possibility: in a directed graph one has two functions $s,t: E \to V$ called source and target. So one could up this to three functions! Then any selection of 2 of these would define a directed graph.2012-08-13
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    The relevant term is "hypergraph", which have a large literature. Edges in a hypergraph may be an arbitrary subset of the vertex set.2012-08-19

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