Graphs

a graph is a mathematical structure for representing relationships

a graph consists of a set of nodes (or vertices) connected by edges (or arcs)
some graphs are directed: edges are from one node to another node. directed graphs are also called digraphs.
other graphs are undirected: edges are to indicate if two nodes are connected; no sense of direction
there are some graphs called mixed graphs that allow for both types of edges, but they’re fairly uncommon and we won’t talk about them.
unless specified otherwise, “graph” means “undirected graph”

Formalizing Graphs

to define a graph, we need to specify what the nodes are, and which edges are in the graph.
nodes can be pretty much anything; what about edges?
- an unordered pair is a set $\{ a,b \}$ of two elements $a \neq b.$ (sets are unordered)
  - e.g. $\{0,1\} = \{1,0\}$
an undirected graph is an ordered pair $G=(V,E)$, where
- $V$ is a set of nodes, which can be anything, and
- $E$ is a set of edges, which are unordered pairs of nodes drawn from $V$.
a directed graph is exactly the same, except $E$ is a set of ordered pairs of nodes in $V$.
- Note: a “self-loop”, i.e. an edge from a node to itself, is possible in a digraph but not in an undirected graph.

Lectures 00-05 and topics covered in PS1-PS2.
You can bring a double-sided, 8.5” by 11” sheet of notes with you to the exam, decorated however you’d like.
- recommendation: hand write it
you can revise and resubmit, new score is max of old score and 85% of your new score.
first midterm is in STLC 111 (for last name A-L), Monday, October 24th, 7-10PM
extra practice problems, a set of 24 practice problems on the topics covered by the upcoming midterm
- many are just old midterm questions!