9.2: Graph Terminology

Special Simple Graphs Complete GraphsK 1,… CyclesC 3,… WheelsW 3,… N-cubesQ 1,… Complete bipartiteK 2,2,…

Special Graphs (see Fig01)

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

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

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.

Examples of degree bcddeg(a)= deg(b)= adeg(c)= deg(d)= efgdeg(e)= deg(f)= deg(g)=

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

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?

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

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.

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

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 abc Deg - (a) Deg + (a) Deg - (b) Deg + (b) def Deg - (c) Deg + (c) Deg - (d) Deg + (d) Deg - (e) Deg + (e) Deg - (f) Deg + (f)

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.

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.

Which of the examples are bipartite? Q: Which of the examples of the worksheet are bipartite? Cycles, complete graphs C3C4C5C6 (see Fig01)

Is this graph bipartite? (see gr_th_ex1) bac g fde bac g fde

Is this graph bipartite? (see gr_th_ex2) ab fc ed ab fc ed

Complete Bipartite Graphs K m,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:

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

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)