Simple Graphs Degrees, Isomorphism, Paths

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1 Mathematics for Computer Science MIT 6.042J/18.062J Simple Graphs Degrees, Isomorphism, Types of Graphs Simple Graph this week Multi-Graph Directed Graph next week Albert R Meyer, March 10, 2010 lec 6W.1 Albert R Meyer, March 10, 2010 lec 6W.2 A simple graph: A Simple Graph edge Definition: A simple graph G consists of a nonempty set, V, of vertices a set, E, of edges edge = set of 2 vertices) vertices, V undirected edges, E ::= {, } adjacent Albert R Meyer, March 10, 2010 lec 6W.3 Albert R Meyer, March 10, 2010 lec 6W.5 Vertex degree degree of a vertex is # of incident edges deg ) = 2 Vertex degree degree of a vertex is # of incident edges deg ) = 4 Albert R Meyer, March 10, 2010 lec 6W.9 Albert R Meyer, March 10, 2010 lec 6W.10 1

2 Possible Impossible Graph? Graph Is there a graph with vertex degrees 2,2,1? Handshaking Lemma sum of degrees is twice # edges NO! 2 2 orphaned edge 1 Proof: Each edge contributes 2 to the sum on the right Albert R Meyer, March 10, 2010 lec 6W.11 Albert R Meyer, March 10, 2010 lec 6W.12 Handshaking Lemma sum of degrees is twice # edges = odd, so impossible Albert R Meyer, March 10, 2010 lec 6W.13 Sex in America: Men more Promiscuous? Study claims: Men average many more partners than women. Graph theory shows this is nonsense Albert R Meyer, March 10, 2010 lec 6W.14 Sex Partner Graph Counting pairs of partners M F now divide by both sides by M partners avg-degm) avg-degf) Albert R Meyer, March 10, 2010 lec 6W.15 Albert R Meyer, March 10, 2010 lec 6W.16 2

3 Average number of partners Averages differ solely by ratio of females to males. No big difference Nothing to do with promiscuity picture of a graph Albert R Meyer, March 10, 2010 lec 6W.17 Albert R Meyer, March 10, 2010 lec 6W.20 picture of same graph picture of same graph Albert R Meyer, March 10, 2010 lec 6W.21 Albert R Meyer, March 10, 2010 lec 6W.22 picture of same graph All that matters are the connections: graphs with the same connections are isomorphic Albert R Meyer, March 10, 2010 lec 6W.23 Albert R Meyer, March 10, 2010 lec 6W.28 3

4 Isomorphism two graphs are isomorphic when there is an edge-preserving matching of their vertices. Albert R Meyer, March 10, 2010 lec 6W.29 Dog Are these isomorphic? Pig Cow Cat Beef Tuna fdog) = Beef fcat) = Tuna Hay Corn fcow) = Hay fpig) = Corn Albert R Meyer, March 10, 2010 lec 6W.30 Edges preserved? Edges preserved? YES! Dog Pig Hay Dog Pig Hay Corn Corn Cow Cat Beef Tuna Cow Cat Beef Tuna Albert R Meyer, March 10, 2010 lec 6W.31 Albert R Meyer, March 10, 2010 lec 6W.32 Nonedges preserved? YES! Formal Def of Graph Isomorphism Dog Pig Hay G 1 isomorphic to G 2 means edge-preserving vertex matching: Corn Cow Cat Beef Tuna bijection f:v 1 V 2 with u v in E 1 iff fu) fv) in E 2 isomorphic! Albert R Meyer, March 10, 2010 lec 6W.34 Albert R Meyer, March 10, 2010 lec 6W.35 4

5 Nonisomorphism degree 2 all degree 3 Proving nonisomorphism If some property preserved by isomorphism differs for two graphs, then they re not isomorphic: # of nodes, # of edges, degree distributions,. Albert R Meyer, March 10, 2010 lec 6W.36 March 10, 2010 lec 6W.37 Finding an isomorphism? Are these two graphs isomorphic? many possible mappings: large search can use properties preserved by isomorphisms as a guide, for example: a deg 4 vertex adjacent to a deg 3 can only match with a deg 4 vertex also adjacent to a deg 3 but even so...nothing known is sure to be much faster than searching thru all bijections for an isomorphism Albert R Meyer, March 10, 2010 lec 6W.38 March 10, 2010 lec 6W.39 Source unknown. All rights reserved. This content is excluded from our Creative Commons license. For more information, see Albert R Meyer, March 10, 2010 lec 6W.41 Albert R Meyer, March 10, 2010 lec 6W.42 5

6 Albert R Meyer, March 10, 2010 lec 6W.43 Albert R Meyer, March 10, 2010 lec 6W.44 ) Albert R Meyer, March 10, 2010 lec 6W.45 Albert R Meyer, March 10, 2010 lec 6W.46 Connectedness two vertices are connected iff there is a path from one to the other. a graph is connected iff every two vertices are connected. Simple Simple Path: all vertices different ) Albert R Meyer, March 10, 2010 lec 6W.47 Albert R Meyer, March 10, 2010 lec 6W.48 6

7 Simple Simple Path: doesn t cross itself) ) & Simple Lemma: The shortest path between two vertices is simple! Proof: by contradiction) suppose path from u to v crossed itself: u c v Albert R Meyer, March 10, 2010 lec 6W.49 Albert R Meyer, March 10, 2010 lec 6W.50 & Simple Lemma: The shortest path between two vertices is simple! then Proof: path by contradiction) without c---c suppose is path shorter! from u to v crossed itself: u c v Team Problems Problems 1 3 Albert R Meyer, March 10, 2010 lec 6W.51 Albert R Meyer, March 10, 2010 lec 6W.53 7

8 MIT OpenCourseWare J / J Mathematics for Computer Science Spring 2010 For information about citing these materials or our Terms of Use, visit:

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