Department of Mathematics, Statistics
and Computer Science
Wim Ruitenburg's Fall 2014 MATH 1300-101
Last updated: November 2014
Comments and suggestions: Email wimr@mscs.mu.edu
Networks from chapter 7
First some terminology for graphs.
- Our graphs have finitely many nodes or vertices.
- Our graphs have finitely many links or edges.
- Most of our graphs are connected, and so consist of just one
component.
Disconnected graphs have multiple components, where even a single isolated node
is considered a component.
- Nodes have a degree of separation, how many links does one need to
follow to get from one to the other?
- A weighted network is one where the links are labeled, usually by
positive numbers.
- What is a circuit?
- What is a bridge?
- What is a tree?
- What is the all-bridges property?
- What is the N-1 link property?
- What is the redundancy of a graph?
We consider graphs from 2 different perspectives.
- For unlabeled graphs we look for minimal subgraphs that
span the collection of all nodes.
- We find 2 ways to built such minimal spanning trees.
First method: Remove edges from circuits.
Second method: Restore necessary edges between nodes as long as you strictly
have to.
- Minimal spanning trees have one edge less than they have nodes.
- In case we have a weighted connected graph, we have Kruskal's
algorithm with which to find a cheapest minimal spanning tree.
Example Problem(s)
- Recommended problems from Chapter 7 of the book:
3, 7, 9, 11
- Recommended problems from Chapter 7 of the book:
21, 25
- Recommended problems from Chapter 7 of the book:
33, 36