Discrrete mathematics for computer science simple graph warmup

Simple Graph Warmup


Cycles in Simple Graphs

•
•

A cycle in a simple graph is a sequence of vertices v0, …, vn for
some n>0, where v0, ….vn-1 are distinct, v0=vn, and {vi,vi+1}
is an edge for 0≤iFact: There can be no cycles of length 0, 1, or 2. (Why?)


Trees

•
•

A tree is a connected acyclic graph.
(These are a kind of undirected graph, so the definition is
different from the directed trees we talked about earlier.)


In a tree there is a unique path between any pair
of vertices

•
•

•

•

Proof. If there were two vertices with no path between them, the
graph would not be connected.
Suppose there were two vertices with two distinct paths between
them. By the WOP there would be a shortest possible path in the
graph for which there is a distinct path between the same pair of
endpoints. The second path must have no vertices in common with
the first except the endpoints (otherwise first path would not be the
shortest).
Then following one path forward and the other back creates a cycle,
contradiction.
So there cannot be more or less than one path between any pair of
vertices. QED.


Finis