1. aka “Undirected Graphs”
Simple Graphs
aka “Undirected Graphs”
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2. Graph Types of Graphs Simple Directed Graph Multi-Graph
before spring break
Multi-Graph
this week
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3. A simple graph G consists of a nonempty set, V, of vertices, and
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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4. vertices, V undirected edges, E edge A Simple Graph “adjacent ”
::= { , }
“adjacent ”
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5. V={a,b,c,d,e,f} E={{a,d},{a,e},{b,c},{b,e}, {b,f},{c,f},{d,f},{e,f}}
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}} b d a e f c
picture of G
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6. deg( ) = 2 degree of a vertex is # of incident edges Vertex degree
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7. deg( ) = 4 degree of a vertex is # of incident edges Vertex degree
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8. NO! Is there a graph with vertex degrees 2,2,1? Impossible Graph
orphaned edge
NO! 2 1
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9. sum of degrees is twice # edges 2 to the sum on the right
Handshaking Lemma
sum of degrees is
twice # edges
Proof: Each edge contributes
2 to the sum on the right
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10. 2+2+1 = odd, so impossible sum of degrees is twice # edges
Handshaking Lemma
sum of degrees is
twice # edges
2+2+1 = odd,
so impossible
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11. Sex in America: Men more Promiscuous?
Study claims:
Men average many more
partners than women.
/03/how-many-sex-partners-do-people-really-have/
Graph theory shows
this is nonsense
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12. Sex Partner Graph
partners M F
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13. Counting pairs of partners
divide by both sides by |M|
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14. Averages differ solely by ratio of females to males. No big difference
Average number of partners
1.035
Averages differ solely by
ratio of females to males.
No big difference
Nothing to do with promiscuity
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15. Some Special Graphs
Complete graph Kn: A graph with n vertices including all possible edges
K5:
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16. Bipartite Graph
Graph in which vertices fall into two disjoint subsets and all edges have one endpoint in each
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17. Isomorphic Graphs d b c a f e
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 a 2 b 4 c 6 d 3 e 1 f 5 1 2 b d a e f c 3 4 5 6
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18. Finis
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