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lec0206 - Undirected Graphs Graphs are composed of two...

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Undirected Graphs Graphs are composed of two components: vertices and edges. Vertices are essentially points. (They are also referred to as nodes.) Typically, they will be labeled on a graph. In an undirected graph, edges are simply lines in between pairs of vertices. So, for example, in a graph with n vertices, the maximum number of edges is n C 2 = n(n-1)/2. This is the number of edges in a complete graph. A complete graph is a graph where there exists an edge between all pairs of vertices. We will define the degree of each vertex of a graph to be the number of edges that are incident to that vertex. A walk in a graph is a sequence of edges that can be traversed one by one. (This essentially means that the endpoint of an edge in a path has to be the starting point of the next edge in the path.) It is permissible for a walk to start and end in the same place. (Or, of course, start and end in different places.) A graph is connected if there exists a path in between all pairs of vertices. Here is an example of a undirected graph:
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Directed Graphs The only difference between a directed graph and an undirected one is how the edges are defined. In an undirected graph, an edge is simply defined by the two vertices it connects. But, in a directed graph, the “direction” of the edge matters. For example, let’s say a graph has two vertices v and w. In a directed graph with these two vertices, we would be allowed to have more than one edge. We could have an edge from v TO w, and another one from w TO v. In essence, each edge not only connects a pair of vertices, but also has a direction associated with it. Using a formal definition, the set of edges of a directed graph G can be defined as a subset of the Cartesian product V x V, where V is the set of vertices. All of the other definitions listed for undirected graphs apply to directed ones. Also, when we talk about the degree of a vertex in a directed graph, we have to distinguish between the “in” degree and the “out” degree. At each vertex, edges are either coming in or going out. The in degree of a vertex is simply the sum of the number of edge coming in to that vertex. The out degree is defined similarly.
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