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Understanding Key Concepts in Graph Theory: Terminology and Special Graphs

This article delves into essential graph theory concepts, focusing on terminology and special graphs, including complete graphs, cycles, and bipartite graphs. It explains the degree of vertices, the Handshaking theorem, and directed graphs, providing insights into their properties. The document also covers classic examples like N-cubes and various graph structures, enhancing your understanding of undirected and directed graphs. Ideal for students and enthusiasts, this guide clarifies fundamental principles in graph theory and equips readers with the tools to analyze graph structures effectively.

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Understanding Key Concepts in Graph Theory: Terminology and Special Graphs

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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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