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9.2: Graph Terminology

9.2: Graph Terminology. Special Simple Graphs. Complete Graphs K 1 ,… Cycles C 3 ,… Wheels W 3 ,… N-cubes Q 1 ,… Complete bipartite K 2,2 ,…. Special Graphs (see Fig01). complete bipartite: K 2,3 and K 3,3. N-cubes: Q1, Q2, Q3, and Q4 (see Fig02).

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9.2: Graph Terminology

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  1. 9.2: Graph Terminology

  2. Special Simple Graphs • Complete Graphs K1,… • Cycles C3,… • Wheels W3,… • N-cubes Q1,… • Complete bipartite K2,2,…

  3. Special Graphs (see Fig01)

  4. complete bipartite: K 2,3 and K 3,3

  5. N-cubes: Q1, Q2, Q3, and Q4 (see Fig02)

  6. Basic Terminology – Undirected Graphs Def: If e={u,v} is an edge, u and v are adjacent. The edge e is incident with vertices u and v. e connects u and v. The degree of a vertex v, deg(v), is the number of edges incident with it, with loops contributing twice.

  7. Examples of degree b c d deg(a)= deg(b)= a deg(c)= deg(d)= e f g deg(e)= deg(f)= deg(g)=

  8. Theorem 1: The Handshaking Theorem: • Let G=(V,E) be an undirected graph with e edges. • Then = ____

  9. Questions Example: How many edges are there in a graph with 10 vertices each of degree 6? Question: Could you construct a graph with 1 vertex of odd degree?

  10. Questions Could you construct a graph:  With 2 vertices of odd degree? With 3, 4, 5,… vertices of odd degree?

  11. Thm. 2: Theorem 2: An undirected graph has an even number of vertices of odd degree. Proof idea: Let V1 be the set of vertices of odd degree and V2 be the set of vertices of even degree in the undirected graph G=(V,E). Then, using Thm. 1, ___= = + … Therefore, there are an even # of vertices of odd degree. 

  12. Directed Graphs- Basic Terms Terms If (u,v) is an edge, u is adjacent to v, and v is adjacent from u u is the initial vertex, and v is the terminal vertex

  13. Deg- (v) and Deg+ (v) – Def and Ex Deg- (v) is the in degree of v: the number of edges with v an the terminal vertex Deg + (v) is the out degree of v: the number of edges with v as the initial vertex inout a b c Deg- (a) Deg + (a) Deg- (b) Deg + (b) d e f Deg- (c) Deg + (c) Deg- (d) Deg + (d) Deg- (e) Deg + (e) Deg- (f) Deg + (f)

  14. Thm. 3 Theorem 3: Let G=(V,E) be a graph with directed edges Then = ______ Def: The underlying undirected graph is the undirected graph that results from ignoring directions of edges on a directed graph.

  15. Bipartite  Def: A simple graph G is called bipartite if its vertex set V can be partitioned into disjoint nonempty sets V1 and V2 such that: If there is an edge between 2 vertices, then one vertex is an element of V1 and one vertex is an element of V2.

  16. Which of the examples are bipartite? Q: Which of the examples of the worksheet are bipartite? Cycles, complete graphs C3 C4 C5 C6 (see Fig01)

  17. Is this graph bipartite? (see gr_th_ex1) b a c g f d e

  18. Is this graph bipartite? (see gr_th_ex2) a b f c e d

  19. Complete Bipartite Graphs Km,n is the graph that is partitioned into two subsets V1 and V2 of m and n vertices where There is an edge between two vertices iff one vertex is in V1 and the other is in V2. Examples:

  20. Local Area Networks • Star Topology, Ring Topology, Hybrid • Parallel Processing v. Serial

  21. New graphs from old • Def: A subgraph of G=(V,E) is a graph H=(W,F) where WV and F E. • Def: The union of two simple graphs G1=(V1,E1) and G2=(V2,E2) is the simple graph G1 G2=( V1  V2, E1  E2)

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