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 ...

The adjacency graphs of a complex

Graph Theory and Q-Analysis

This paper is a collection of thoughts and observations, being partly a review and partly a report of current research, on recent work in various aspects of Grunbaum colorings, their existence and usage. In particular, one of the most striking significances of Grunbaum's Conjecture in the 2-dimensional case is its equivalence to the 4-Color Theorem. The notion of Grunbaum coloring is extended from the 2-dimensional case to the case of arbitrary finite hyper-dimensions.

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Grunbaum coloring and its generalization to arbitrary dimension

ABSTRAcr. The (r, s)-adjacency graph of a simplicial complex K has been defined as the graph whose nodes are the r-cells of K with adjacency whenever there is incidence with a common s-cell. The (r, s)-adjacency graphs for r > s have been characterized by graph coverings by Dewdney and Harary generalizing the result of Krausz for line-graphs (r = 1, s = 0). We now complete the characterization by handling the case r < s.

/pdf/characterization-of-r-s-adjacency-graphs-of-complexes-2e8d4t622v.pdf

Characterization of $(r,\,s)$-adjacency graphs of complexes

/pdf/the-adjacency-graphs-of-a-complex-41sb1qp9ce.pdf

Graph theory

: A survey of the known results on graphs and complexes associated as incidence patterns to different types of graphs, complexes, or families of sets. Among the topics discussed are interchange graphs, interval graphs, total graphs, and nerves. (Author)

The adjacency graphs of a complex

Citations

Graph Theory and Q-Analysis

Grunbaum coloring and its generalization to arbitrary dimension

Characterization of $(r,\,s)$-adjacency graphs of complexes

References

Graph theory

Incidence patterns of graphs and complexes

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