Chapter 12

Examples from Graph Theory: An Introduction to the Lucas Numbers

Let us start with some basic ideas from graph theory.

An undirected graph G is made up of a nonempty set V of vertices and a set E (possibly empty) of edges between the vertices in V. For example, in Fig. 12.1 (a), we have img and E = {uv, vw, vv}, where it is understood, for example, that uv = vu. Here we say that img, img are neighbors and u, img are also neighbors—but u, img are not neighbors. Also, the edge vv shown in the figure is an example of a loop (at img).

In Fig. 12.1 (b), we have a second undirected graph—this time img and E = {uv, vw, ux, xy, yu}. Here the edges ux, xy, and yu constitute a cycle. Since this cycle ...

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