By Jurisic A.

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Extra resources for 1-Homogeneous Graphs with Cocktail Party, mu -Graphs

Sample text

On the other hand, suppose that no vertex of G has eccentricity 1. This means that for every vertex u of G, there is some vertex v of G such that uv ∈ E(G). Now, let H be a new graph, constructed by adding a single vertex, w, to G, together with the edges {wx | x ∈ V (G)}. In the graph H, the eccentricity of w is 1 (w is adjacent to everything). Further, for any vertex x ∈ V (G), the eccentricity of x in H is 2 (no vertex of G is adjacent to everything else in G, and everything in G is adjacent to w).

3 Trees w Y w N y? x Y N z? w x? Y N y Y y? Y z? N x Y 33 N z? y N z? Y N w is a z is a y is a z is a x is a z is a y is a z is a max. max. max. max. max. max. max. max. 33. Logic of a program. 34. A few saturated hydrocarbons. UNC Memphis Wash. St. UNC Louisville Louisville UNC Tennessee Kansas Kansas Memphis Kansas Memphis Mich. St. Texas Stanford Memphis Texas UCLA UCLA Villanova Kansas UCLA Wisconsin Davidson Xavier W. Kentucky Xavier W. 35. The 2008 Men’s Sweet 16. Exercises 1. Draw all unlabeled trees of order 7.

For the reverse direction, suppose a graph G of order n is connected and contains n − 1 edges. We need to show that G is acyclic. If G did have a cycle, we could remove an edge from the cycle and the resulting graph would still be connected. In fact, we could keep removing edges (one at a time) from existing cycles, each time maintaining connectivity. The resulting graph would be connected and acyclic and would thus be a tree. 10. Therefore, G has no cycles, so G is a tree. 13. A graph of order n is a tree if and only if it is acyclic and contains n − 1 edges.

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1-Homogeneous Graphs with Cocktail Party, mu -Graphs by Jurisic A.


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