Discrete Structures

# Lecture Notes, 3 Dec 2010

 Last Update: 3 December 2010 Note: or material is highlighted

### §8.2 (pp. 541–542) and §9.1: Graphs, including Digraph Representation of Binary Relations

1. Graphs:

1. Defs:

• A vertex (or: a node) ≈def a geometric point
• An edge (or: an arc) ≈def a line connecting 2 vertices.

• Let V be a non-∅ set (of vertices).
Let E be a set of edges.
Then:

1. G is a graph =def G = (V,E)

• i.e., a graph is an ordered pair consisting of
a set of vertices and a set of edges

• ∴ a graph is just a certain kind of set
(in fact, a binary relation!)

2. Let G=(V,E) be a graph.
Let v1,v2 ∈ V.
Let e ∈ E.
Then:

v1,v2 are e's endpoints
and e connects v1 and v2
=def
e ≈ {v1,v2}

• Notes:

1. It would be wrong to say "e={v1,v2}",
because, ∀v1,v2 ∈ V,
there can be ≥0 e ∈ E;
i.e., there can be more than one edge connecting any pair of vertices

2. More precisely:

V can be any set whatsoever.
E is typically a "bag" or "multiset"
(a set with duplicates)
of (unordered) pairs of members of V

2. E.g.)

3. Defs:

Let G=(V,E) be a graph.
Then:

1. G is a di(rected) graph =def E ⊆ V×V

• i.e.) E is a (multi-)set of ordered pairs of vertices

2. e = (v1,v2) ∈ E
(e isdef a directed edge
e's initial vertex =def v1
e's terminal vertex =def v2)

3. e isdef a loop =def e = (v,v) ∈ E

• E.g.)

2. Binary relations can be represented by (or "as") digraphs: