By Philippe Mathis

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**Additional info for Graphs and Networks: Multilevel Modeling**

**Sample text**

Similarly, in certain, even very simple, cases the dual is not of the same type as the primal in the sense that a primal 1-graph can have a p-graph as a dual, as we will see. This may present a problem in terms of graph description, since the adjacency matrix of the dual then has to be an extremely hollow p-dimensional matrix… 16 For example, the definition of the dual graph of a grid (adjacency matrix) provided by Pumain will not be retained here: [PUM 97], page 31. xxxii Graphs and Networks The concept of dual graph is very rich, but insufficiently used.

Urban planners need representations that are repetitive and verifiable, as well as comprehensible for all the public, in particular, within the framework of public interest investigations. Moreover, we have seen that graph theory is absolutely not preoccupied with the representation in the sense of graphic plotting of a graph. In a certain manner Berge eliminates the problem by writing: “it is only necessary to know how the nodes are connected. The localization of the nodes in the figure, the representation or plotting of the graph do not count” [BER 70, BER 87].

128-140, Opera Omnia, vol. 7, p. 1-10, 1736. , “Demonstreatio nonnullarum insignium proprietatum quibus solidra hedris planis inclusa sunt praedita”, Novi Comm. Acad. Sci. Imp. Petropl. 4 (1752-1753, published in 1758), 140-160, Opera Omnia (1), vol. 26, 94-108, 1758. , Elementa doctrina solidorum. Novi Comm Acad. SCI. Imp. Petropol. 4 (1752-1753, published in 1758), 109-140, Opera Omnia (1) vol. 26, 72-93, 1758. , Eléments de recherche opérationnelle, Collection “Programmation”, Gauthier-Villars, Paris, 1975.