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Unformatted text preview: AMS/MBA 546 (Fall, 2010) Estie Arkin Network Flows: Solution sketch to homework set # 1 1). Recall the definiton: A sequence S = d 1 ,d 2 ,...,d n is graphical if there exists an undirected (simple) graph with n nodes such that the degree of node i is equal to d i . A simple graph is a graph with no loops and no multiple edges. (a). Show that if a sequence is graphical, and n 2, then there must be two numbers d i , d j such that d i = d j . (In other words, graphical sequences cannot have all their elements be unique.) In a simple graph on n nodes, each node has degree 0 , 1 ,...,n 1. In order for each node to have a different degree (there are n nodes and n different possible degrees) we must have a node of degree 0, a node of degree 1, etc, including a node of degree n 1. The node of degree n 1 must therefore be adjacent to all other nodes in the graph, but that would mean no node has degree 0. (b). Let S 1 = d 1 ,d 2 ,...,d n and S 2 = w 1 ,w 2 ,...,w n where w i = n 1 d i for i = 1 ,...,n . Show that S 1 is graphical if and only if S 2 is graphical....
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This note was uploaded on 01/31/2011 for the course AMS 546 taught by Professor Arkin,e during the Fall '08 term at SUNY Stony Brook.
- Fall '08