Types of Graphs Mathematics for Computer Science MIT 6.042J/18.062J Simple Directed Graph Graph Simple Graphs: Degrees last week Multi-Graph this week degrees.1 degrees.2 Albert R Meyer April 1, 2013 Albert R Meyer April 1, 2013 A simple graph: A Simple Graph edge vertices, V Definition: A simple graph G consists of undirected edges, E •a nonempty set, V, of vertices, and ::= { , } •a set, E, of edges such that each edge has two endpoints in V “adjacent ” degrees.3 degrees.4 Albert R Meyer April 1, 2013 Albert R Meyer April 1, 2013 1
Vertex degree Vertex degree degree of a vertex is degree of a vertex is # of incident edges # of incident edges deg( ) = 2 deg( ) = 4 degrees.6 degrees.7 Albert R Meyer April 1, 2013 Albert R Meyer April 1, 2013 Handshaking Lemma Possible Graph? Impossible Graph sum of degrees is Is there a graph with twice # edges vertex degrees 2,2,1? ∑ 2|E | = deg(v) ? orphaned edge NO! v V ∈ 2 1 Proof : Each edge contributes 2 2 to the sum on the right degrees.8 degrees.9 Albert R Meyer April 1, 2013 Albert R Meyer April 1, 2013 2
Handshaking Lemma Sex in America: Men more Promiscuous? • Univ Chicago, The Social Organization sum of degrees is of Sexuality , 1994: men have 74% more twice # edges women partners than women have men • NBC News, American Sex Survey , 2004: ∑ 2|E | = deg(v) men have 233% more women partners v V ∈ • U.S. National Center for Health Statistics study, 2007: men have 175% more women 2+2+1 = odd, partners so impossible degrees.10 sexgraph.11 Albert R Meyer April 1, 2013 Albert R Meyer April 1, 2013 Sex in America: Men more Promiscuous? Sex Partner Graph Studies claim different %’s but agree that M F men average many more partners than women. Graph theory shows this is nonsense partners sexgraph.12 degrees.13 Albert R Meyer April 1, 2013 Albert R Meyer April 1, 2013 3
Counting pairs of partners Counting pairs of partners ∑ deg(m) ∑ deg(m) avg degree(M) ::= m ∈ M m ∈ M M M avg degree(M) avg degree(F) = ∑ ∑ deg(f) deg(f) f ∈ F f ∈ F avg degree(F) ::= F F ∑ ∑ deg(m) = E = deg(f) m ∈ M f ∈ F degrees.14 degrees.16 Albert R Meyer April 1, 2013 Albert R Meyer April 1, 2013 Average number of partners Counting pairs of partners F ⋅ avg-deg(M) avg-deg( ) F = 1.035 M avg degree(F) = F avg degree(M) Averages differ solely by ratio of females to males. M No big difference Nothing to do with promiscuity degrees.17 degrees.19 Albert R Meyer April 1, 2013 Albert R Meyer April 1, 2013 4
Why are surveys wrong? Maybe people are lying: • Males exaggerate? • Females deny? Maybe Males have partners outside the study sexgraph.20 Albert R Meyer April 1, 2013 5
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