Solution to Width of a tree is defined as the largest distance(number of edges in the shortest … - Sikademy
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Archangel Macsika

Width of a tree is defined as the largest distance(number of edges in the shortest path) between any two vertices. What is the smallest possible width of the tree with 100 vertices? 1 2 99 101 1 point How many complete graphs with at least 1 edge and at most 50 edges are there? 8 9 10 11

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Here's the Solution to this Question

(1) Consider a graph called star. This graph is a special kind of tree with one internal node and k leaves and has the smallest possible width. A star S_{99} has 99 edges and 100 vertices and its width is 2.

Answer: 2.


(2) The complete graph K_n on vertices has n(n-1)\over2. In order to answer that question, we need to solve equation for positive integer n:


1\le {n(n-1)\over2} \le 502\le n^2 - n \le 100

n \in \{2, 3,4,5,6,7,8,9,10\}

Therefore, there are 9 complete graphs with at least 1 edge and at most 50 edges (K_2, K_3, \dots, K_{10}).

Answer: 9 graphs.


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