1. 9.4 Connectivity

2. Invariant properties
Recall- invariant properties that isomorphic graphs share Some examples?... Path lengths are another invariant property. Applications of paths: sending a message between any 2 computers, taking a bus from a to b

3. Definitions of path, circuit
Basic Def:
In a simple undirected graph, a path of length n from x0 to xn is a sequence x0, x1,…xn. 
A circuit is a path where x0=xn.
A simple path or circuit is one that does not contain the same edge more than once.
In an undirected graph: A path of length n from u to v: A sequence of edges e1,e2,…en such that f(e1)={ x0, x1}…f(en)= {xn-1, xn} where x0=u and xn=v.
In a directed graph, the notation changes: f(e1)=(x0, x1)…f(en)= (xn-1, xn).

4. Path- examples
Hollywood
Collaboration
See book p

5. Connectedness in Undirected Graphs
 Def: An undirected graph is called connected if there is a path between every pair of distinct vertices on the graph.
Examples and non-examples

6. Connected, unconnected
In this connected graph, removal of which edges would make the graph unconnected? A d f g B c e h i

7. Thm. 1
Theorem 1: There is a simple path between every pair of distinct vertices of a connected undirected graph. Proof method?

8. Thm 1 proof
Proof: Let u and v be two distinct vertices of the connected undirected graph G=(V,E). Since ____________, there is at least one path between u and v. Let x0,x1,…xn be a path from u to v _________. To see it is simple, assume_________ Then…

9. Connectedness in Directed Graphs
 Def: A directed graph is strongly connected if there is a path from a to b and from b to a whenever a and b are vertices in the graph.
Def: A directed graph is weakly connected if there is a path between any two vertices in the underlying graph.
Question: Does one imply the other?

10. Ex-- strongly and weakly connected
a b c
d e f
Ex 2

11. Paths and Isomorphism Look for paths on the handout from 9.3.
  If you can find a path or circuit of a certain length in G but not in H, then G and H are not isomorphic.
Another example: consider vertices, edges, degrees,… paths
1 1  

12. Counting Path Between Vertices
 Ex: How many paths of length 3 are there from a to d in G?
a b
d c
Find the adjacency matrices for A, A2, A3

13. Counting paths- Ex. 2
Ex 2: How many paths of length 4 are there from a to b?
a b
d c
Find A and A4

14. Theorem 2:
Thm. 2: Let G be a graph with an adjacency matrix A. The number of different paths of length r from vi to vj equals the (i,j)th entry of Ar.
Proof: method? … see book for proof

15. Exercises
See p. 631: see sketchpad