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Disc-a11 - no edges and H to be the subgraph of G that...

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1. A graph has 7 vertices , with degrees 0 , 1 , 2 , 2 , 3 , 4 , and 4 . (a) How many edges does the graph have ? 0 + 1 + 2 + 2 + 3 + 4 + 4 2 = 8 (b) Draw an example of such a graph . One such graph is this e e e e e e e ° ° ° ° @ @ @ @ 2. Let G be a graph with 10 vertices and 15 edges . (a) How many induced subgraphs does G have ? You get an induced subgraph by taking a subset of the vertices, so there are 2 10 of them. (b) How many spanning subgraphs does G have ? You get a spanning subgraph by taking a subset of the edges, so there are 2 15 of them. 3. Construct a graph G and a subgraph H of G so that ° ( H ) < ° ( G ) . What are the numbers ° ( H ) and ° ( G ) for your example ? The simplest example is to take G to be a graph with two vertices and
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Unformatted text preview: no edges, and H to be the subgraph of G that consists of one of those vertices. Then & ( H ) = 1 < 2 = & ( G ) . 4. Construct a graph G and a subgraph H of G so that & ( H ) > & ( G ) and ! ( H ) < ! ( G ) . What are the numbers & ( H ) , & ( G ) and ! ( H ) for your example ? The simplest example is to take G to be a graph with two vertices and one edge between them, and H to be the subgraph of G with the same vertices but no edges. Then & ( H ) = 2 > 1 = & ( G ) while ! ( H ) = 1 < 2 = ! ( G ) ....
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