f2006lecNotesWeek10

# f2006lecNotesWeek10 - 27 27.1 Introduction to Graphs Graph...

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27 Introduction to Graphs 27.1 Graph basics a graph is a set of the form G =( V,E ) V is a set of vertices (or nodes ) E is a set of edges representation: draw a picture •• v 1 v 2 v 3 v 4 e 1 e 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V = { v 1 ,v 2 3 4 } edges can be denoted by e i or by set of endpoints { v j k } E = { e 1 ,e 2 } where e 1 = { v 2 3 } 2 = { v 3 4 } if { v i j }∈ E we say that v i and v j are adjacent in the example, v 2 and v 3 are adjacent as are v 3 and v 4 variations in types of graphs based on forms of edges can an edge connect a vertex to itself to form a loop? can we have more than one edge connecting two vertices? two-way vs one-way edges digraphs we have already seen these in our study of relations draw a picture v 1 v 2 v 3 v 4 e 1 e 2 e 3 e 4 e 5 . . . . . . . . . . . . . . . . . . . . . .......... . . . . . . . . . . . . . . . . . . . . . . . ................................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ±± ² ³ ´ V = { v 1 2 3 4 } E = { e 1 2 3 4 5 } where e 1 v 1 1 ) 2 v 2 2 ) 3 v 2 3 ) 4 v 3 2 ) 5 v 4 3 ) note that edges are ordered pairs, not sets with digraphs, adjacent to and adjacent from in the example, v 4 is adjacent from v 3 while v 3 is adjacent to v 4 simple graphs no loops, at most one edge adjacent to two vertices draw a picture multigraphs can have more than one edge connecting the same pair of vertices such edges are called parallel edges draw a picture pseudographs now can also have loops draw a picture directed multigraphs digraphs with parallel edges draw a picture 58

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27.2 Some special simple graphs complete : K n pictures “Komplett” cycles : C n ( n> 2) pictures with variations on pictures for C 4 , square and bow tie •• ................................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . for C 5 , pentagon and pentagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........... ....................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....................... ........... wheels : W n ( 2) pictures note: W n has n + 1 vertices n-cubes : Q n ( 0) each vertex in Q n is of degree n pictures of Q 1 ,Q 2 3 note: Q n has 2 n vertices note relationship to bit strings of length n edges connect strings that diﬀer by one bit 27.3 Bipartite graphs deFnition: a bipartite graph is a simple graph in which all vertices can be placed in one of two sets with no adjacent vertices in either set illustration . . . .
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## This note was uploaded on 04/19/2008 for the course ECE 190 taught by Professor Carter during the Fall '06 term at University of Toronto.

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f2006lecNotesWeek10 - 27 27.1 Introduction to Graphs Graph...

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