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2. Find the leaf on Ti with the smallest label and call it v. 3. Record in the sequence the label of v’s neighbor. 4. Remove v from Ti to create a new tree Ti+1 . 5. If Ti+1 = K2 , then stop. Otherwise, increment i by 1 and go back to step 2. Let us run through this algorithm with a particular graph. 47, tree T = T0 has 7 vertices, labeled as shown. The first step is finding the leaf with smallest label: This would be 2. The neighbor of vertex 2 is the vertex labeled 4. Therefore, 4 is the first entry in the sequence.

Iii. If every vertex of G is marked, then the set of marked edges forms a minimum weight spanning tree. If not, repeat step ii. 44. As you work, compare the stages to those of Kruskal’s algorithm. 7. Give an example of a connected, weighted graph G having (i) a cycle with two identical weights, which is neither the smallest nor the largest weight in the graph, and (ii) a unique minimum weight spanning tree which contains exactly one of these two identical weights. 4 Counting Trees As for everything else, so for a mathematical theory: beauty can be perceived but not explained.

If two vertices are in different components, however, we say that the distance between them is infinity. We conclude this section with two interesting results. Choose your favorite graph. It can be large or small, dense with edges or sparse. Choose anything you like, as long as it is your favorite. Now, wouldn’t it be neat if there existed a graph in which your favorite graph was the “center” of attention? The next theorem (credited to Hedetneimi in [44]) makes your wish come true. 5. Every graph is (isomorphic to) the center of some graph.

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