   Chapter 10.4, Problem 4ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Let G be the graph of a hydrocarbon molecule with the maximum number of hydrogen atoms for the number of its carbon atoms. a. Draw the graph of G if G has three carbon atoms and eight hydrogen atoms. b. Draw the graphs of three isomers of C 5 H 12 . c. Use Example 10.4.4 and exercise 3 to prove that if the vertices of G Consist of k carbon atoms and m hydrogen atoms, then G has a total degree of 2 k + 2 m − 2 .d. Prove that if the vertices of G consist of k carbon atoms and m hydrogen atoms, then G has a total degree of 4 k + m .e. Equate the results of (c) and (d) to prove Cayley’s result that a saturated hydrocarbon molecule with k carbon atoms and a maximum number of hydrogen atoms has 2 k + 2 hydrogen atoms.

To determine

(a)

To draw:

The graph of G if G has three carbon atoms and eight hydrogen atoms.

Explanation

Given information:

Let G be the graph of a hydrocarbon molecule with the maximum number of hydrogen atoms for the number of its carbon atoms.

Calculation:

G is the graph of a hydrocarbon molecule with the maximum number of hydrogen atoms for the number of carbon atoms.

G has 3 carbon atoms and 8 hydrogen atoms.

The 3 carbon atoms are represented by 3 vertices and all carbon atoms need to be connected by an edge...

To determine

(b)

To draw:

The graph of three isomers of C5H12.

To determine

(c)

To prove:

If the vertices of G consist of k carbon atoms and m hydrogen atoms, then G has a total degree of 2k + 2m − 2.

To determine

(d)

To prove:

If the vertices of G consist of k carbon atoms and m hydrogen atoms, then G has a total degree of 4k + m.

To determine

(e)

To find:

Equate the results of (c) and (d) to prove Cayley’s result that a saturated hydrocarbon molecule with k carbon atoms and a maximum number of hydrogen atoms has 2k + 2 hydrogen atoms.

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