Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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WHY (A)? EXPLAIN ME PLEASE STEP BY STEP 

3. In chemistry, a fullerene is a roughly spherical shaped collection of carbon atoms arranged in
pentagonal and hexagonal rings. One example is the so-called "Buckminsterfullerne", or "Bucky-
ball"1, shown here in an image grabbed from Wikipedia (you may also recognize the shape of the
Buckminsterfullerene as showing up in a soccer ball).
From our point of view, we can think of a fullerene as a planar graph having the following two
properties:
• Every region of the graph has degree either 5 or 6
• Every vertex of the graph has degree 3
(The region degree restriction shows up for chemical reasons - smaller rings of carbon atoms are
too unstable. The vertex degree restriction comes from how if you had 4 pentagons or hexagons at
a single point, they would add up to more than 360 degrees).
Part a: Suppose that a Fullerene graph has exactly P pentagons and H hexagons. In terms of P,
and H, how many edges does the Fullerene graph have? How many vertices does it have?
SOL.
(a) P + H - edges + vertices = 2 (Euler's formula)
5P + 6H
edges
Each edge is shared by two faces, thus number of edges
5P + 6H
edges
And since 3 faces meet to form a vertex, number of vertices
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Transcribed Image Text:3. In chemistry, a fullerene is a roughly spherical shaped collection of carbon atoms arranged in pentagonal and hexagonal rings. One example is the so-called "Buckminsterfullerne", or "Bucky- ball"1, shown here in an image grabbed from Wikipedia (you may also recognize the shape of the Buckminsterfullerene as showing up in a soccer ball). From our point of view, we can think of a fullerene as a planar graph having the following two properties: • Every region of the graph has degree either 5 or 6 • Every vertex of the graph has degree 3 (The region degree restriction shows up for chemical reasons - smaller rings of carbon atoms are too unstable. The vertex degree restriction comes from how if you had 4 pentagons or hexagons at a single point, they would add up to more than 360 degrees). Part a: Suppose that a Fullerene graph has exactly P pentagons and H hexagons. In terms of P, and H, how many edges does the Fullerene graph have? How many vertices does it have? SOL. (a) P + H - edges + vertices = 2 (Euler's formula) 5P + 6H edges Each edge is shared by two faces, thus number of edges 5P + 6H edges And since 3 faces meet to form a vertex, number of vertices
Expert Solution
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Step 1

Handshaking Theorem: This theorem says that, in any graph the sum of degree of all vertices is equal to the twice of number of edges in that graph.

Euler's Formula: The relation between number of vertices v, number of edges e and number of faces F of a graph is                                                                     v-e+F=2

Consider a Fullerene graph with exactly P pentagon and H hexagon. Therefore total number of Faces are F=P+H.

Let total number of vertices in the Fullerene graph are v and total number of edges are e. Since degree of each vertices in Fullerene graph is 3, therefore sum of degree of all vertices is 3v, therefore applying Handshaking Theorem:

                                                            vdegVertex=2ev3=2e3v=2ev=2e3

 

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