Discrrete mathematics for computer science graphs

Simple Graphs aka “Undirected Graphs” 3/19/12 Types of Graphs Simple Directed Graph Graph before spring break this week 3/19/12 Multi-Graph 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) 3/19/12 A Simple Graph edge vertices, V undirected edges, E ::= { , } “adjacent ” 3/19/12 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 3/19/12 picture of G Vertex degree degree of a vertex is # of incident edges deg( ) = 3/19/12 Vertex degree degree of a vertex is # of incident edges deg( ) = 3/19/12 PossibleGraph Graph? Impossible Is there a graph with vertex degrees 2,2,1? ? NO! orphaned edge 2 3/19/12 Handshaking Lemma sum of degrees is twice # edges 2| E | = ∑ deg(v) v∈ V Proof: Each edge contributes to the sum on the right 3/19/12 Handshaking Lemma sum of degrees is twice # edges 2| E | = ∑ deg(v) v∈ V 2+2+1 = odd, so impossible 3/19/12 10 Sex in America: Men more Promiscuous? Study claims: Men average many more partners than women Graph theory shows this is nonsense 3/19/12 http://drjengunter.wordpress.com/2011/12 /03/how-many-sex-partners-do-people-really-have/ 11 Sex Partner Graph M F partners 3/19/12 12 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) 3/19/12 = F M ∑ deg(f ) × f ∈F F     avg-deg(F) 13 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 with promiscuity 3/19/12 14 Some Special Graphs • Complete graph Kn: A graph with n vertices including all possible edges K5: 3/19/12 15 Bipartite Graph • Graph in which vertices fall into two disjoint subsets and all edges have one endpoint in each 3/19/12 16 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 d b c a f e 3/19/12 a b c d e f 17 Finis 3/19/12 18 ... each 3/19/12 16 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...Types of Graphs Simple Directed Graph Graph before spring break this week 3/19/12 Multi-Graph A simple graph: Definition: A... ratio of females to males No big difference Nothing to with promiscuity 3/19/12 14 Some Special Graphs • Complete graph Kn: A graph with n vertices including all possible edges K5: 3/19/12 15