M122Graphs - Graph Theory The Definition of a Graph A graph...

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Unformatted text preview: Graph Theory The Definition of a Graph A graph is an ordered pair G = ( V, E ), where V is a finite, non-empty set of objects called vertices , and E is a (possibly empty) set of unordered pairs of distinct vertices ( i.e. , 2-subsets of V ) called edges . The set V (or V ( G ) to emphasize that it belongs to the graph G ) is called the vertex set of G , and E (or E ( G ) to emphasize as above) is called the edge set of G . If e = { u, v } E ( G ), we say that vertices u and v are adjacent in G , and that e joins u and v . Well also say that u and v are the ends or endpoints of e . The edge e is said to be incident with u (and v ), and vice versa. We write uv (or vu ) to denote the edge { u, v } , on the understanding that no order is implied. Notes: (1) E ( G ) is a set. This means that two vertices either are adjacent or are not adjacent. There is no possibility of more than one edge joining a pair of vertices. (2) The elements of E are 2-subsets of V . Thus a vertex can not be adjacent to itself. There are more general definitions in which there may be more than one edge between two vertices, and/or vertices may be adjacent to themselves. Sometimes these go by the names multigraphs or pseudographs . Sometimes authors intend this more general situation when they say graphs, and the objects we have defined above are then usually called simple graphs . Representing Graphs Pictorially By definition, a graph is a pair of sets. Graphs are usually represented pictorially with a point (or dot) in the plane corresponding to each vertex and a line segment (or curve of some sort) joining the corresponding points for each pair of adjacent vertices. The picture tells you what the graph is, that is, it tells you what the vertices are, and what the edges are. The same graph can have many different pictorial representations. How the picture is drawn is unimportant. It is the information that the picture represents that is important. Walks, Trails, Paths, and Cycles A v- v n walk in a graph is an alternating sequence of vertices and edges, v , e 1 , v 1 , e 2 , v 2 , e 3 , v 3 , . . . , e n , v n such that e i = v i- 1 v i for 1 i n . The integer n is the length of the walk. It is the number of edges in the walk, one less than the number of vertices. You can think of a walk in a graph G as the result of the process of starting at a vertex of G , physically walking along the edges of G (no turning around half way!), and recording the names of the vertices encountered (including the start and end), and the edges used. Thus the length equals the number of edges youd travel over. According to our definition of a graph there is at most one edge between any two vertices. Thus, recording the edges in a walk amounts to recording redundant information. (However, in a more general situation 1 where there can be more than one edge between two vertices it would be important to record which edge was used.) In our situation, an equivalent definition of a walk...
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M122Graphs - Graph Theory The Definition of a Graph A graph...

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