This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 }...
View
Full
Document
This note was uploaded on 03/14/2012 for the course MTHSC 810 taught by Professor Staff during the Fall '08 term at Clemson.
 Fall '08
 Staff
 Math

Click to edit the document details