Applications of graph theory to group structure by Claude Flament

By Claude Flament

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Extra resources for Applications of graph theory to group structure (Prentice-Hall series in mathematical analysis of social behavior)

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20, where the paths, the tracks, and the deviations can immediately be seen. We shall present first a method of graphic marking, and second, a matrix method (using the matrix associated with the graph). The marking method to compute the deviations from one point to all others. In a graph G = (X; r), to obtain the deviations from the point a E X to all the points of X, we proceed according to the following steps: 1. We mark as 0 (zero) the point a. 2. We mark as k the points not already marked and which are part of (rkx) 3.

We have seen that an articulation point in a graph G is a point whose removal from G makes the graph of a certain type of connectivity an unconnected one. Ross and Harary (1959) suggest considering each point of a graph to see what its removal entails for the graph's type of connectivity: the removal of certain points may lower its connectivity, while the removal of others may raise it. These are weakening or strengthening points. Hence the notion of the power of a point. An articulation point is obviously a weakening point.

This distance between two graphs G; = (X; V,) and G; = (X; Vj) can therefore be defined as follows: d(G;Gj)=IV;DV;I. n by the union and the intersection of their sets of arcs : Gi u Gj = (X; V1 u Vj); G,nGj=(X;V;nV;). The same notions can be applied to 2°n if the term arc is replaced by edge in the above discussion. Notice that the distance d(GQG;) in 2 is not equal to the distance d(GG;) in Y,,. 48 9. Various Types of Graphs Up to now we have considered graphs which represent a correspondence F within a set X; this allows us only to represent a relation in an all-or-none fashion between, say, the members of a group.

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