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Discrete Mathematics 5. Graphs & Trees
Typology: Lecture notes
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Discrete Mathematics, Spring 2009
-^ Technical meaning in discrete mathematics:^ A particular class of discrete structures (to bedefined) that is useful for representing relations andhas a convenient webby-looking graphicalrepresentation.
a set
V^ of
vertices
or^
nodes
corresponds to the
Visual Representationof a Simple Graph
Discrete Mathematics, Spring 2009
−^ a set
V^ of
vertices
or^
nodes
corresponds to the
universe of the relation
R), and
−^ a set
E^ of
edges
/^ arcs
/^ links
: unordered pairs of
[distinct?]
elements
u,v
, such that
uRv
-^ Let
-^ Let
Discrete Mathematics, Spring 2009
-^ Definition
A^ pseudograph G
f^ ) where
f:E→
{{u,
v}| u,v
}. Edge
e∈ E^ is a
loop
if
f(e)={
u,u
}={u
-^ Example:
nodes are campsitesin a state park and edges arehiking trails through the woods.
loop
Discrete Mathematics, Spring 2009
-^ Definition
A^ directed graph
E) consists of a set of vertices
V^ and a binary relation
E^ on
-^ Example:
V^ = people,
x,y) |
x^ loves
y}
-^ Like directed graphs, but there may be more than onearc from a node to another. •^ Definition
A^ directed
multigraph
,^ f^ ) consists of a set
V^ of
Discrete Mathematics, Spring 2009
A^ directed
multigraph
,^ f^ ) consists of a set
V^ of
vertices, a set
E^ of edges, and a function
f:E
-^ Example
−^ The WWW is a directed multigraph. −^ V
=web pages,
E=hyperlinks.
-^ Keep in mind this terminology is not fully standardized...
Discrete Mathematics, Spring 2009
T er m
E d g ety p e
M u ltip leed g es o k?
S elf-lo o p s o k?
S im p le g rap h
U n d ir.
N o
N o
M u ltig rap h
U n d ir.
Y es
N o
P seu d o g rap h
U n d ir.
Y es
Y es
D irected g rap h
D irected
N o
Y es
D irected m u ltig rap h
D irected
Y es
Y es
-^ u
Discrete Mathematics, Spring 2009
-^ Edge
-^ Edge
-^ Vertices
-^ The
Discrete Mathematics, Spring 2009
-^ A vertex of degree 1 is
u^ is
adjacent to
v,^ v
is^ adjacent from
u
Discrete Mathematics, Spring 2009
−^ u
is^ adjacent to
v,^ v
is^ adjacent from
u
−^ e comes from
u, e
goes to
v.
−^ e connects u to v
,^ e goes from u to v
−^ the
initial vertex
of^
e^ is
u
−^ the
terminal vertex
of^
e^ is
v
Let
G^ be a directed graph,
v^ a vertex of
−^ The
in-degree
of^
v, deg
−(v), is the number of
edges going to
v. Discrete Mathematics, Spring 2009
edges going to
v.
−^ The
out-degree
of^
v, deg
+(v), is the number of
edges coming from
v.
−^ The
degree
of^
v, deg(
v)≡
deg
−(v)+deg
+(v), is the
sum of
v’s in-degree and out-degree.
-^ Cycles
Discrete Mathematics, Spring 2009
-^ Cycles
-^ Wheels
n
-^ n
-^ Bipartite Graphs •^ Complete Bipartite Graphs
,n
−^ For any
n∈
N, a
complete graph
on
n^ vertices,
K,n
is a simple graph with
n^ nodes in which every
node is adjacent to every other node:
∀u
,v∈
Discrete Mathematics, Spring 2009
node is adjacent to every other node:
∀u
,v∈
u≠v
u,v
) (^1) ( 2 (^11)
− − n^ nn = i ∑= i