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Chapter 11: Problem 36

Is there a nonempty simple graph with twice as many edges as vertices? Explain. (You may find it helpful to use the result of exercise 34.)

Short Answer

Expert verified
In conclusion, there is no nonempty simple graph with twice as many edges as vertices. Using the Handshaking Theorem and the fact that the sum of degrees of all vertices should be an even number, we can conclude that such a graph cannot exist due to the limitations on the number of edges and the relationship between edges and vertices.

Step by step solution

01

Understand the Handshaking Theorem

The Handshaking Theorem states that the sum of the degrees of all vertices in a graph is equal to twice the number of edges or \(\sum_{i=1}^{n} d(v_i) = 2e\), where \(d(v_i)\) is the degree of vertex \(i\), \(n\) is the number of vertices, and \(e\) is the number of edges. The theorem gets its name due to its analogous application in a scenario where each handshake between persons can be considered an edge, and the number of handshakes one person has is their degree.
02

Apply the Handshaking Theorem Constraint

Since we are looking for a graph with twice as many edges as vertices, we can represent this condition as \(e = 2n\). Now we can express the Handshaking Theorem with this condition: \(\sum_{i=1}^{n} d(v_i) = 2(2n) = 4n\).
03

Observe Parity of Degrees

Since a graph's vertices' degrees are all integers, the sum of degrees will always be an even number. If we divide both sides of the equation by 2, we get \(\frac{1}{2} \sum_{i=1}^{n} d(v_i) = 2n\). This shows that the sum of the degrees of all vertices in our desired graph should be an even number.
04

Check the Number of Vertices

To have twice the number of edges as vertices, we must have at least two vertices in the graph. Let's consider the simple graph with two vertices (n=2). The relationship between edges and vertices would be \(e = 2n = 4\). However, in a simple graph with only two vertices, there can only be a single edge between them, which does not satisfy our condition.
05

Conclude the Existence

In conclusion, there is no nonempty simple graph with twice as many edges as vertices. The Handshaking Theorem and the limitation on the number of edges prevent such a graph from existing.

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Most popular questions from this chapter

Find graphs that have the following adjacency matrices. a. \(\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 2 \\ 1 & 2 & 0\end{array}\right] \quad\) b. \(\left[\begin{array}{lll}0 & 2 & 0 \\ 2 & 1 & 0 \\\ 0 & 0 & 1\end{array}\right]\)

Show that at a party with at least two people, there are at least two mutual acquaintances or at least two mutual strangers.

Graph with four vertices of degrees \(1,2,3\), and \(3 .\)

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