sol2 - AMS 301.3 Fall, 2009 Homework Set # 2 Solution Notes...

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AMS 301.3 Fall, 2009 Homework Set # 2 — Solution Notes # 4, Supplement II: A graph with n vertices can have at most ( n 2 ) = n ( n - 1) 2 vertices; this would be the case of a complete graph, K n , in which all n vertices have degree ( n - 1). Thus, if we have 3 vertices, there are at most 3 edges, if we have 4 vertices, there are at most 6 edges, etc. If we have n = 10, there are at most ( 10 2 ) = 45 edges, so it is impossible to have a 10-vertex graph with 50 edges. (NOTE: We are assuming here that it is a simple graph, not a multigraph; in a multigraph, even if there is only ONE vertex, it is possible to have 50 (or 50,000) edges — just make them all self-loops!) If we have n = 11, there are at most ( 11 2 ) = 55 edges (and K 11 has exactly 55 edges). Thus, with 11 vertices it is possible to have 50 edges (e.g., take K 11 and delete any 5 edges), and it is not possible to have 50 edges if there are fewer than 11 vertices. #6, 1.3: The number of edges in
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sol2 - AMS 301.3 Fall, 2009 Homework Set # 2 Solution Notes...

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