An Example
The graph in which vertices 1 through n are placed in a row with each vertex connected by an unlabelled edge to its immediate neighbors gives rise to the symmetric group Sn+1; the generators correspond to the transpositions (1 2), (2 3), ... (n n+1). Two non-consecutive transpositions always commute, while (k k+1) (k+1 k+2) gives the 3-cycle (k k+2 k+1). Of course this only shows that Sn+1 is a quotient group of the Coxeter group described by the graph, but it is not too difficult to check that equality holds.
Read more about this topic: Coxeter Group
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