9.4 Connectivity

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

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).

Path- examples Hollywood Collaboration See book p. 623-624

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

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

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

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…

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?

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

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 6 2 6 2 5 3 5 3   4 4

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

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

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

Exercises See p. 631: 19-21 see sketchpad