Discrrete mathematics for computer science graphs
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Simple Graphs
aka “Undirected Graphs”
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Types of Graphs
Simple
Directed Graph
Graph
before spring break
this week
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Multi-Graph
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A simple graph:
Definition:
A simple graph G consists of
a nonempty set, V, of vertices, and
a set, E, of edges such that
each edge has two distinct
endpoints in V
Write G = (V,E)
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A Simple Graph
edge
vertices, V
undirected edges, E
::= { , }
“adjacent ”
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A simple graph G:
V={a,b,c,d,e,f}
E={{a,d},{a,e},{b,c},{b,e},
{b,f},{c,f},{d,f},{e,f}}
d
b
c
a
f
e
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picture of G
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Vertex degree
degree of a vertex is
# of incident edges
deg( ) = 2
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Vertex degree
degree of a vertex is
# of incident edges
deg( ) = 4
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PossibleGraph
Graph?
Impossible
Is there a graph with
vertex degrees 2,2,1?
?
NO!
orphaned edge
2
1
2
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Handshaking Lemma
sum of degrees is
twice # edges
2| E | = ∑ deg(v)
v∈ V
Proof:
Each edge contributes
2 to the sum on the right
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Handshaking Lemma
sum of degrees is
twice # edges
2| E | = ∑ deg(v)
v∈ V
2+2+1 = odd,
so impossible
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Sex in America: Men more Promiscuous?
Study claims:
Men average many more
partners than women.
Graph theory shows
this is nonsense
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/>
/03/how-many-sex-partners-do-people-really-have/
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Sex Partner Graph
M
F
partners
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Counting pairs of partners
∑ deg(m) =E =∑ deg(f )
m∈M
f ∈F
divide by both sides by |M|
∑ deg(m)
m∈M
M
avg-deg(M)
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=
F
M
∑ deg(f )
×
f ∈F
F
avg-deg(F)
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Average number of partners
F
avg- deg(M) = 1.035 ×avg- deg(F)
M
Averages differ solely by
ratio of females to males.
No big difference
Nothing to do with promiscuity
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Some Special Graphs
•
Complete graph Kn: A graph with n vertices including all possible
edges
K5:
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Bipartite Graph
•
Graph in which vertices fall into two disjoint subsets and all
edges have one endpoint in each
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Isomorphic Graphs
•
Graphs (V,E) and (V’,E’) such that there is a bijection f: V→V’ that
preserves edges: {v,w}∈E iff {f(v),f(w)}∈E’
•
Any two complete graphs of the same size are isomorphic
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d
b
5
c
a
f
e
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6
4
3
a
2
b
4
c
6
d
3
e
1
f
5
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Finis
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