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Chapter 12: Problem 10

The connected undirected graph \(G=(V, E)\) has 30 edges. What is the maximum value that \(|V|\) can have?

Short Answer

Expert verified
The maximum number of vertices \(|V|\) that the graph can have is 8.

Step by step solution

01

Recall the maximum number of edges in a graph

In an undirected graph, the maximum number of edges \(E\) is given by the formula \(E = \frac{|V|*(|V|-1)}{2}\).
02

Adjust the formula to reflect the problem scenario

We know the number of edges \(E\), which is 30. So, we substitute \(E\) with 30 in the formula, to get: \(30 = \frac{|V|*(|V|-1)}{2}\).
03

Solve the formula for \(|V|\)

Since it's an algebraic equation, it can be solved in several ways. The important thing is to isolate \(|V|\) on one side. This typically involves simplifying the equation, then applying the quadratic formula or completing the square. Once we solve this quadratic equation, we find \(|V|\) = 8 or \(|V|\) = -7. Since \(|V|\) can't be negative in a graph, the maximum number of vertices \(|V|\) that the graph can have is 8.

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Most popular questions from this chapter

How many leaves does a full binary tree have if its height is, (a) \(3 ?\) (b) \(7 ?\) (c) \(12 ?\) (d) \(h\) ?

Let \(G=(V, E)\) be a loop-free connected undirected graph with \(v \in V\) a) Prove that \(\overline{G-v}=\bar{G}-v\). b) If \(v\) is an articulation point of \(G\), prove that \(v\) cannot be an articulation point of \(\bar{G}\).

On the first Sunday of 2003 Rizzo and Frenchie start a chain letter, each of them sending five letters (to ten different friends between them). Each person receiving the letter is to send five copies to five new people on the Sunday following the letter's arrival. After the first seven Sundays have passed, what is the total number of chain letters that have been mailed? How many were mailed on the last three Sundays?

Give an example of an undirected graph \(G=(V, E)\) where \(|V|=|E|+1\) but \(G\) is not a tree.

Consider the complete binary trees on 31 vertices. (Here we distinguish left from right as in Example 12.9.) How many of these trees have 11 vertices in the left subtree of the root? How many have 21 vertices in the right subtree of the root?
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