- Your next homework is due Tuesday.
- The withdrawal date is coming up. If you'd like to talk to me about your progress before that date, please make an appointment to see me or stop by during my office hours.
- Today we continue talking about graphs.
- Graphs:
- The Bridges
of Konigsberg problem: "is it possible to set off and walk around Konigsberg crossing each bridge exactly one?" (p. 116)
- Solved by Euler
(1735), the beginning of graph theory.
- Graphs are made up of vertices (points) and edges
that connect the vertices. More formally, a graph is
- A collection of points, called vertices.
- A collection of edges, each of which connects two vertices.
- The degree of a vertex is the number of edges coming into it.
- Graphs are made up of vertices (points) and edges
that connect the vertices. More formally, a graph is
- Euler's solution:
- Konigsberg: To be traversable, there must be at most two vertices of odd degree.
- Euler noticed the hand-shaking theorem: in any graph, the number of vertices with odd degree must be even.
- An alternative solution -- thanks to spiked math
- Which of the Platonic solid graphs can be traversed without backtracking? (here they are)
- Solved by Euler
(1735), the beginning of graph theory.
- Examples of graphs:
- The Platonic solid graphs
- Graphs of the dual solids from the Platonic solid graphs
- Trees
- Graph that are
- weighted: each edge (connection between vertices) has a number attached to it:
- labelled:
- The graph above has vertices labelled.
- Sometimes the edges are labelled too.
- Sometimes a label indicates a state of being (e.g. LAX); sometimes a label indicates the result of an action (e.g. a coin toss).
- Examples:
- Fibonacci's Bunny Tree
- Hexstat Probability generator
- weighted: each edge (connection between vertices) has a number attached to it:
- Graphs don't change by bending edges, but breaking them or
detaching edges from their vertices (and hence creating new vertices)
gives new graphs.
- Two graphs are the same (isomorphic) if one can be
relabel the vertices of one and relabel the edges so that the graphs
can be drawn identically.
Hence, for example,
- the graph of the tetrahedron and the Mercedes Benz symbol are isomorphic as graphs
- the graphs of the regular pentagons, star and convex, are isomorphic.
- A graph is simple if it doesn't have any loops (edges connected
from a vertix to itself), or multiple edges connecting the same two
vertices (as there are in the Konigsberg graph).
Exercise: draw all the distinct simple graphs with three vertices.
- A cycle is a route from a node back to itself that doesn't retrace steps.
- One of these graphs is a tree: a graph that has no cycles.
- Duality -- again! The dual of a graph is obtained by
- throwing out the connections that exist between vertices, and
- making any connections that don't exist.
We can use duality to save ourselves some work.
Exercise: draw all the simple graphs with four vertices.
- Have you ever seen any of these before? Could we give any of them names?
Here are all simple graphs with five vertices -- there are 34!
- Planar graphs are graphs that can be drawn such that no two edges
intersect. Every Platonic
graph can be drawn this way, so they are planar graphs.
- Complete graphs are simply graphs with connections between
every pair of vertices (but no loops).
- Exercise: Let's see if we can show that the complete graph
with five vertices (
)is not planar:
- We can show that the complete graph with four vertices is planar.
- Why not? What happens?
- The three utilities and three houses graph is not planar:
- Any non-planar graph has a copy of one of these two graphs
in it somewhere, as a subgraph:
- The Bridges
of Konigsberg problem: "is it possible to set off and walk around Konigsberg crossing each bridge exactly one?" (p. 116)
