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Earl C F, Johnson J H, 1981, "Graph theory and *Q*-analysis" *Environment and Planning B* **8**(4) 367 – 391

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**Graph theory and Q-analysis**

**C F Earl, J H Johnson**

Received 9 November 1981; in revised form 29 November 1981

**Abstract. **Structures of graph theory are compared with those of q-analysis and there are many similarities. The graph and simplicial complex defined by a relation are equivalent in terms of the information they represent, so that the choice between graph theory and *q*-analysis depends on which gives the most natural and complete description of a system. The higher dimensional graphs are shown to be simplicial families or complexes. Although network theory is very successful in those physical science applications for which it was developed, it is argued that *Q*-analysis gives a better description of human network systems as patterns of traffic on a backcloth of simplicial complexes. The *q*-nearness graph represents the *q*-nearness of pairs of simplices for a given *q*-value. It is concluded that known results from graph theory could be applied to the *q*-nearness graph to assist in the investigation of *q*-connectivity, to introduce the notion of connection defined by graph cuts, and to assist in computation. The application of the *q*-nearness graph to *q*-transmission and shomotopy is investigated.

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