Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

Representation: For graphs without multiple edges we can use adjacency lists or
matrices. For general graphs we can use incidence matrices.
Definition: Let G(V,E) have no multiple edges. The adjacency list L G = {a v } v,V ,
where a v = adj(v) = {w,V | w is adjacent to v}.
Definition: Let G(V,E) have no multiple edges. The adjacency matrix A G = [a ij ]
is
!
a ij
= 1 if {v i ,v # } is an edge of G,
j
"
# 0 otherwise.
Example:
$
v 1
v 2
v 4
v 3
results in A G
=
0 1 1 0
1 0 0 1
1 0 0 1
0 1 1 0
!
#
#
#
#
#
#
"#
$
&
&
&
&
&
&
%&
and L G
=
v 1
:
v 2
:
v 3
:
v 4
:
!
#
#
"
#
#
#
$
v 2
,v 3
v 1
,v 4
v 1
,v
.
4
v 2
,v 3
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Note: For an undirected graph, A G
= A G T . However, this is not necessarily true
for a directed graph.
Definition: The incidence matrix M = [m ij ] for G(V,E) is
!
m ij
= 1 when edge e i is incident with v j ,
#
"
# 0 otherwise.
$
Definition: The simple graphs G(V,E) and H = (W,F) are isomorphic if there is
an isomorphism f: V%W, a one to one, onto function, such that a and b are
adjacent in G if and only if f(a) and f(b) are adjacent in H for all a,b,V.
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