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PPT – Lecture 09: Basic Graph Theory PowerPoint presentation | free to download - id: 150632-OTBlM

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Lecture 09 Basic Graph Theory

- Introduction to Graphs
- Basic Definitions
- Complete Graph/Cycle
- Bipartite Graphs
- Operations on Graphs
- Other Graph Representations
- Graph Isomorphism

9.1. Basic Definitions

- A graph G (V, E)
- Is a discrete structure consisting of
- A nonempty set of vertices (nodes) V and
- a set of edges E that connect these vertices.
- First consider graph that contains
- No arrows (directions)
- No loops edges from a vertex to itself.
- No multiple edges joining the same two vertices.
- Also referred to as undirected simple graph.

9.1. Basic Definitions

- A graph can be described either by the use of
- Sets or
- a diagram.
- The diagram can be drawn differently and still

represents the same graph. - Example
- Sets A graph with vertices V ? A, B, C, D ?

and edges E ? ?A,B?, ?B,C?, ?C,D? ?.

9.1. Basic Definitions

- Adjacent, incident, and degree
- Two vertices u and v in V are adjacent or

neighbors if there is an edge e between u and v. - The edge e ?u, v? joins (or connects/links) the

vertices u and v. - The vertices u and v are endpoints of e.
- The edge e is incident with the vertex u and v.
- The degree of u, denoted deg(u), is the number of

edges incident with a vertex u.

9.1. Basic Definitions

- Example
- Vertices A and B are adjacent.
- Vertices A and D are not because there is no edge

between them. - deg(A) 1
- deg(B) 3
- deg(C) 0, C is called isolated.

9.1. Basic Definitions

- The Handshaking Theorem
- Let G (V, E). Then
- Proof
- Each edge contributes twice to the degree count

of all vertices. (Q.E.D.) - Example
- If a graph has 3 vertices, can each vertex have

degree 3? Or 4? - The sum is 3?3 9 which is an odd number. Not

possible. - The sum is 4?3 12 2 E and 12/2 6 edges.

May be possible.

9.2. Some Special Graphs

- Complete graphs - Kn
- The (simple) graph with
- n vertices
- exactly one edge between every pair of distinct

vertices. - Examples
- K5 is important because it is the simplest

nonplanar graph. It cannot be drawn in a plane

with nonintersecting edges.

9.2. Some Special Graphs

- Complete graphs - Kn
- How many edges of Kn?

9.2. Some Special Graphs

- Cycles Cn
- Is an n vertex graph with all vertices having a

degree two, and all are connected. - Examples

9.3. Bipartite Graphs

- A bipartite graph G is
- Is a graph in which the vertices V can be

partitioned into two disjoint subsets V1 and V2

such that - No two vertices in V1 are adjacent.
- No two vertices in V2 are adjacent.

9.3. Bipartite Graphs

- Examples
- Let the courses and professors be the vertices of

a graph G. - Put an edge between a course and a professor

whenever the professor can teach the course. - The resulting graph is bipartite.

9.3. Bipartite Graphs

- Examples
- Is a Star network (K1,n) bipartite graph?
- Is a cycle Ck bipartite graph?

9.3. Bipartite Graphs

- Examples
- Is the following graph bipartite?
- Solution
- If a is in V1 then b, d and e must be in V2

(why?) - Then c is in V1 and there is no inconsistency.
- The graph is a complete bipartite graph K2,3.

9.3. Bipartite Graphs

- A bipartite graph G is
- Is a graph in which the vertices V can be

partitioned into two disjoint subsets V1 and V2

such that - No two vertices in V1 are adjacent.
- No two vertices in V2 are adjacent.
- A complete bipartite graph Km,n is
- Is a bipartite graph in which the sets V1 and V2

contain m and n vertices, respectively, and every

vertex in V1 is adjacent to every vertex in V2. - How many edges ?

9.4. Operations on Graphs

- A subgraph of G (V, E)
- Is a graph H (W, F) where W ? V and F ? E.
- Examples
- K5 and its subgraph

9.4. Operations on Graphs

- The union of two graphs G1 (V1, E1) and G2

(V2, E2) - Is a graph with vertex set V1 ? V2 and edge set

E1 ? E2. - i.e. G1 ? G2 (V1 ? V2, E1 ? E2)
- Examples
- G1 and G2 with their union G1 ? G2

9.5. Other Graph Representations

- To analyze a graph using a computer, we need to

communicate the vertices and edges of the graph. - Graph representations
- Adjacency list
- Adjacency matrices

9.5. Other Graph Representations

- Adjacency list of a graph G
- Specify the vertices that are adjacent to each

vertex of the graph. - Examples
- A graph and its adjacency list.

9.5. Other Graph Representations

- Adjacency matrix of a graph G (V, E) with n

vertices - Is an n ? n matrix A aij, where
- Sum of the entries in row i of the adjacency

matrix ? - The matrix is symmetric, i.e. aij aji.

9.5. Other Graph Representations

- Examples
- Two graphs and their adjacency matrices.

9.6. Graph Isomorphism

- The two graphs below are in fact the same graph

(structure-wise). - We say that these graphs are isomorphic.

9.6. Graph Isomorphism

- Let G1 (V1, E1) and G2 (V2, E2) be simple

graphs. - The graphs G1 and G2 are isomorphic iff
- There exists a bijection f V1?V2
- The function f is called an isomorphism of G1

with G2. - i.e. For all vertices u and v in V1,
- if u and v are adjacent in G1
- then f(u) and f(v) are adjacent in G2.

9.6. Graph Isomorphism

- Graph isomorphism invariant properties
- Are the properties that G1 and G2 must have in

common in order to be isomorphic - the same number of vertices.
- the same number of edges.
- degrees of corresponding vertices are the same.
- if one is bipartite, the other must be.
- if one is complete, the other must be.
- etc. but these are necessary, not sufficient !

9.6. Graph Isomorphism

- Examples
- The following graphs are isomorphic.

9.6. Graph Isomorphism

- Examples
- The following graphs are not isomorphic. Why?

9.6. Graph Isomorphism

- Examples
- The following graphs are not isomorphic.
- Vertex C in G1 is adjacent to the vertices A, B,

D and E. So it has a degree of 4. - But there are no vertices of degree 4 in G2.

9.6. Graph Isomorphism

- Examples
- Determine if the following two graphs G1 and G2

are isomorphic.

9.6. Graph Isomorphism

- Solution
- Check . . .
- They have the same number of vertices 5
- They have the same number of edges 8
- They have the same number of vertices with the

same degrees 2, 3, 3, 4, 4.

9.6. Graph Isomorphism

- 3?2, 1?1, 5?4, 2?3, 4?5
- The problem in general is very difficult, even

using - a computer !

9.7. Further Readings

- Introduction to Graphs
- Basic Definitions Section 11.1
- Bipartite Graphs p.541
- Graph Isomorphism Section 11.2