9.2: Graph Terminology

Special Simple Graphs

• Complete Graphs K1,…

• Cycles C3,…

• Wheels W3,…

• N-cubes Q1,…

• Complete bipartite K2,2,…

Special Graphs (see Fig01)Complete graphs K n

Cycle graphsC n

W heelsW n

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

Complete bipartite graphs K n,m

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

n-cubes

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

b c d deg(a)=deg(b)=

a deg(c)=deg(d)=

e f g deg(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

TermsIf (u,v) is an edge, u is adjacent to v, and v is

adjacent from uu 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

in outa 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)

Thm. 3

Theorem 3: Let G=(V,E) be a graph with directed edgesThen = ______ 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 graphsC3 C4 C5 C6

(see Fig01)

Is this graph bipartite? (see gr_th_ex1)

b

a c

g

f de

Is this graph bipartite? (see gr_th_ex2)

a b

f c

e d

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:

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)