Mathematics for Computer Science MIT 6.042J/18.062J

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Mathematics for Computer Science MIT 6.042J/18.062J Graphs Copyright © Radhika Nagpal, 2002. Prof. Albert Meyer & Dr. Radhika Nagpal

Connectivity Are two vertices connected? Is there a path from one vertex to another. How many paths are there from u to v …

Example: MIT Buildings Already saw one example last week Can we get from one building to another without going outside? = is the graph one connected component? = is the transitive closure =? VxV

Smallest Connected Graph Question: repeat last part of MIT connected buildings question. Ask how many people did this question? Answer: n-1edges, now we can prove it

Smallest Connected Graph What does this Graph look like? One answer: line Actually any tree ! Lots of choices, could easily optimize for something, e.g. congestion vs shortest distance between nodes (e.g star formation)

Trees Definitions:

In class exercise: isomorphism and connectedness false proof.

Cut Edge Not only concerned with whether soemthing is connected but also how easy is it to disconnect it? Fault-tolerance Definition: An edge is a cut edge if removing it from the graph

Cut-edge In the following graph are A or B cut edges? Anything interesting about A or B that distinguishes them? An edge is a cut edge iff it is not part of a cycle.

Discussion of False Proof Counter Example Problem with False Proof 1 Problem with False Proof 2

Discussion of cut-edge Use this theorem on particular graphs In a tree, every edge is a cut edge In a mesh, no edge is a cut edge

Old slides 1

MIT Building Connections 26 13 12 10 4 8

MIT Building Connections 4 13 10 12 26 8 R 4 13 10 12 26 8 26 13 12 10 4 8

Composition using Matrices 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 1 =

Composition and Path Lengths is the set of all pairs (a,b) such that bldg a and b are connected via < 1 other bldg In general is the set of all pairs (a,b) such that bldg a and b are connected via < k-1 other bldgs. Prove using induction Note: different if edges are not bidirectional