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Unformatted text preview: 2. GRAPH TERMINOLOGY 186 2. Graph Terminology 2.1. Undirected Graphs. Definitions 2.1.1 . Suppose G = ( V,E ) is an undirected graph. (1) Two vertices u,v ∈ V are adjacent or neighbors if there is an edge e between u and v . • The edge e connects u and v . • The vertices u and v are endpoints of e . (2) The degree of a vertex v , denoted deg ( v ) , is the number of edges for which it is an endpoint. A loop contributes twice in an undirected graph. • If deg ( v ) = 0 , then v is called isolated . • If deg ( v ) = 1 , then v is called pendant . Example 2.1.1 . V = { v 1 ,v 2 ,v 3 ,v 4 } and E = { e 1 ,e 2 ,e 3 ,e 4 } . v 1 v 2 v 3 v 4 e 1 e 2 e 3 e 4 (1) v 2 and v 3 are adjacent. (2) deg ( v 1 ) = 2 (3) deg ( v 2 ) = 2 (4) deg ( v 3 ) = 3 (5) deg ( v 4 ) = 1 Discussion Notice that in computing the degree of a vertex in an undirected graph a loop contributes two to the degree. In this example, none of the vertices is isolated, but v 4 is pendant. In particular, the vertex v 1 is not isolated since its degree is 2. 2. GRAPH TERMINOLOGY 187 2.2. The Handshaking Theorem. Theorem 2.2.1 . (The Handshaking Theorem) Let G = ( V,E ) be an undi rected graph. Then 2  E  = X v ∈ V deg ( v ) Proof. Each edge contributes twice to the sum of the degrees of all vertices. Discussion Theorem 2.2.1 is one of the most basic and useful combinatorial formulas associ ated to a graph. It lets us conclude some facts about the numbers of vertices and the possible degrees of the vertices. Notice the immediate corollary. Corollary 2.2.1.1 . The sum of the degrees of the vertices in any graph must be an even number. In other words, it is impossible to create a graph so that the sum of the degrees of its vertices is odd (try it!). 2.3. Example 2.3.1. Example 2.3.1 . Suppose a graph has 5 vertices. Can each vertex have degree 3? degree 4? (1) The sum of the degrees of the vertices would be 3 · 5 if the graph has 5 vertices of degree 3. This is an odd number, though, so this is not possible by the handshaking Theorem. (2) The sum of the degrees of the vertices if there are 5 vertices with degree 4 is 20. Since this is even it is possible for this to equal 2  E  . Discussion If the sum of the degrees of the vertices is an even number then the handshaking theorem is not contradicted. In fact, you can create a graph with any even degree you want if multiple edges are permitted. However, if you add more restrictions it may not be possible. Here are two typical questions the handshaking theorem may help you answer. Exercise 2.3.1 . Is it possible to have a graph S with 5 vertices, each with degree 4, and 8 edges? 2. GRAPH TERMINOLOGY 188 Exercise 2.3.2 . A graph with 21 edges has 7 vertices of degree 1, three of degree 2, seven of degree 3, and the rest of degree 4. How many vertices does it have?...
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 Fall '08
 PenelopeKirby
 Graph Theory, Planar graph, vertices, Qn

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