WHY (A)? EXPLAIN ME PLEASE STEP BY STEP 3. In chemistry, a fullerene is a roughly spherical shaped collection of carbon atoms arranged inpentagonal 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 theBuckminsterfullerene 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 twoproperties:• 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 aretoo unstable. The vertex degree restriction comes from how if you had 4 pentagons or hexagons ata 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 + 6HedgesEach edge is shared by two faces, thus number of edges5P + 6HedgesAnd since 3 faces meet to form a vertex, number of vertices

Answered: Part a: Suppose that a Fullerene graph…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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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

Expert Solution

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 $3 v$ , therefore applying Handshaking Theorem:

$\begin{array}{rcl} \sum_{v} d e g (Vertex) & = & 2 (e) \\ \sum_{v} 3 & = & 2 (e) \\ 3 v & = & 2 e \\ v & = & \frac{2 e}{3} \end{array}$