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Unformatted text preview: MthSc 810: Mathematical Programming Lecture 18 Pietro Belotti Dept. of Mathematical Sciences Clemson University November 29, 2011 Reading for today: Sections 7.1, 7.2 Reading for Thursday: Sections 7.5. Final: take home, out Tue Dec. 6, turn in Tue Dec. 13, 6PM (Undirected) graphs A graph 1 G is a tuple ( V , E ) where V is the set of nodes and E is the set of edges . Each edge is a subset of V of cardinality 2. 1 2 3 4 5 6 V = { 1 , 2 , 3 , 4 , 5 , 6 } , E = {{ 1 , 2 } , { 1 , 4 } , { 1 , 5 } , { 2 , 3 } , { 2 , 4 } , { 3 , 4 } , { 3 , 6 } , { 4 , 5 } , { 5 , 6 }} . The nodes i and j of an edge { i , j } are its endnodes . 1 More precisely, an undirected graph . Directed graphs A directed graph 2 G is a tuple ( V , A ) where V is a set of nodes and A is a set of arcs . Each arc is an ordered pair of nodes in V . 1 2 3 4 5 6 V = { 1 , 2 , 3 , 4 , 5 , 6 } , A = { ( 1 , 4 ) , ( 2 , 3 ) , ( 3 , 2 ) , ( 4 , 2 ) , ( 4 , 6 ) , ( 5 , 4 ) , ( 6 , 5 ) } . In an arc ( i , j ) , i is its tail and j is its head . 2 Or, more simply, digraph Adjacency matrix of a graph Two nodes i and j of a graph G are adjacent if there is an edge { i , j } connecting them The neighborhood of a node i is the set of nodes adjacent to i : N ( i ) = { j V : { i , j } E } The adjacency matrix of a graph G is a (symmetric; why?) matrix M whose elements M ij are 1 if { i , j } E , 0 otherwise: 1 2 3 4 5 6 M = 0 1 0 1 1 0 1 0 1 1 0 0 0 1 0 1 0 1 1 1 1 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 Incidence matrix of a graph The incidence matrix of a graph G is a matrix P with  V  rows and  E  columns, whose element P ij is 1 if the jth edge contains node i , 0 otherwise 1 2 3 4 5 6 M = 1 1 1 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 One row per node, 1 to 6. One column per edge, in the order E = {{ 1 , 2 } , { 1 , 4 } , { 1 , 5 } , { 2 , 3 } , { 2 , 4 } , { 3 , 4 }...
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 Fall '08
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 Math

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