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Degree of Connectivity Within the same type of connectivity, the graphs can be distinguished further. For instance, the two graphs of Fig. 32 are strongly connected, but it is sufficient to delete only one arc in a to change the type of connectivity, while in P it is necessary to delete two. Luce (1950) calls degree of connectivity of a graph the minimal number of arcs that must be deleted in order to make it unconnected. This definition originated within the framework of the study of symmetric graphs, which comprise only one type of connectivity.

Introduction The length of a path y, denoted 1(y), is measured, by definition, by the number of arcs which constitute the path. From a point x to a point y, there may be many paths y,(xy); the deviation from x toy is, by definition, e(xy) = min {l[y,(xy)]}. i This is the length of the shortest path from x to y. A path y(xy) whose length is equal to e(xy) is a track, which is denoted 6(xy). The deviation is not generally a distance. Indeed, when the graph is not symmetric, to go from y to x we are often compelled to pass through points different from those by which we pass to go from x toy; and there is no need for the tracks 6(xy) and 6(yx) to be of the same length.

For instance, the two graphs of Fig. 32 are strongly connected, but it is sufficient to delete only one arc in a to change the type of connectivity, while in P it is necessary to delete two. Luce (1950) calls degree of connectivity of a graph the minimal number of arcs that must be deleted in order to make it unconnected. This definition originated within the framework of the study of symmetric graphs, which comprise only one type of connectivity. In the general framework, we can consider the minimal number of arcs which must be deleted in order for the graph to change from its original type of connectivity to a lower one.

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Applications of graph theory to group structure (Prentice-Hall series in mathematical analysis of social behavior) by Claude Flament


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