aka “Undirected Graphs” Simple Graphs aka “Undirected Graphs” 3/19/12

Graph Types of Graphs Simple Directed Graph Multi-Graph before spring break Multi-Graph this week 3/19/12

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) 3/19/12

vertices, V undirected edges, E edge A Simple Graph “adjacent ” ::= { , } “adjacent ” 3/19/12

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 3/19/12

deg( ) = 2 degree of a vertex is # of incident edges Vertex degree 3/19/12

deg( ) = 4 degree of a vertex is # of incident edges Vertex degree 3/19/12

NO! Is there a graph with vertex degrees 2,2,1? Impossible Graph orphaned edge NO! 2 1 3/19/12

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 3/19/12

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 3/19/12

Sex in America: Men more Promiscuous? Study claims: Men average many more partners than women. http://drjengunter.wordpress.com/2011/12 /03/how-many-sex-partners-do-people-really-have/ Graph theory shows this is nonsense 3/19/12

Sex Partner Graph partners M F 3/19/12

Counting pairs of partners divide by both sides by |M| 3/19/12

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 3/19/12

Some Special Graphs Complete graph Kn: A graph with n vertices including all possible edges K5: 3/19/12

Bipartite Graph Graph in which vertices fall into two disjoint subsets and all edges have one endpoint in each 3/19/12

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 3/19/12

Finis 3/19/12