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)