BasicGraphTheory - ESI 6417 Basic Graph Theory 1 Basic...

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ESI 6417 Basic Graph Theory 1 Basic Graph Theory Ref: AMO Sections 2.1 and 2.2 A graph or network is usually denoted as G ( N , A ) where G is short for ‘graph’ and N and A are names or labels for the sets that contains nodes and arcs in the network. Elements of A are pairs of distinct nodes. When the network is directed, then the order of the nodes in the pair is important, for it indicates the direction. Otherwise (for undirected networks), the order is irrelevant. Example 1: Let G ( N , A ) be a directed network where N = {1, 2, 3, 4} and A = {(1, 2), (1, 3), (2, 3), (3, 2), (2, 4), (3, 4)}. Since the network is directed, the order of nodes in each element of A indicates the direction of the arc. Example 2: Let G ( N , A ) be an undirected network where N = {1, 2, 3, 4} and A = {(1, 2), (1, 3), (3, 2), (2, 4), (4, 3)}. Since the network is undirected, the order of nodes in each element of A is not important Note : In the above, elements of A are pairs of distinct nodes. This is to disallow self- loops or an arc that begins and ends on the same node. For example, the arc (2,2) in the following network is not allowed. 2 14 3 2 1 4 3 1 2
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ESI 6417 Basic Graph Theory 2 In most cases, we will not allow duplicate arcs, e.g., Definition n = the number of elements (nodes) in set N , i.e., n is the cardinality of N . m = the number of elements (arcs) in set A , i.e., m is the cardinality of A . Directed Network Except for the minimum spanning tree problem, we will assume that every network is directed . A directed arc ( i , j ) has two endpoints i and j i = the tail of the arc j = the head of the arc Terminology: We say that 1. Arc ( i , j ) emanates from node i or arc ( i , j ) is an outgoing arc at node i . 2. Arc ( i , j ) terminates at node j or that arc ( i , j ) is an incoming arc at node j . 3. Arc ( i , j ) is incident to nodes i and j . 4. If arc ( i , j ) is in the network, then node j is adjacent to node i . (Note: If the arc ( i , j ) is undirected, j is adjacent to i and i is adjacent to j .) 3 1 2 2 1 3 i j
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ESI 6417 Basic Graph Theory 3 Degree of a node: 1. An indegree of a node is the number of incoming arcs at that node. 2. An outdegree of a node is the number of outgoing arcs at that node. 3. A degree of a node is the sum of indegree and outdegree at that node. Adjacency Lists: 1. An arc adjacency list, A ( i ), of node i is a list of arcs that emanate from node i , i.e., A ( i ) = {( i , j ) A : j N } 2. A node adjacency list, A ( i ), of node i is a list of nodes that adjacent to node i , i.e., A ( i ) = { j N : ( i , j ) A } Example: 2 14 3 Arc Adjacency List Node Adjacency List A (1) = {(1, 2), (1, 4), (1, 3)} A (1) = {2, 3, 4} A (2) = A (2) = A (3) = A (3) = A (4) = A (4) = Note that the arc and node adjacency lists are equivalent, in that you can construct one from the other.
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ESI 6417 Basic Graph Theory 4 Definition: | A ( i )| = # of elements of the set A ( i ) or the cardinality of A ( i ). Note that | A ( i )| is simply the outdegree at node i and we have the following property.
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This note was uploaded on 09/19/2008 for the course ESI 6417 taught by Professor Siriphonglawphongpanich during the Spring '07 term at University of Florida.

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BasicGraphTheory - ESI 6417 Basic Graph Theory 1 Basic...

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