Structure and Properties Of Fullerenes

STRUCTURE AND PROPERTIES OF FULLERENES

Abstract

The fullerenes are allotropes of carbon. The structural formulae for the fullerenes differ from the other allotropes due to the unusual arrangement of the carbon atoms. The layers and the arrangement of the fullerene allotrope can be represented in the algebraic expressions. The algebra involved in the fullerene definitions gives the mathematical representation of the atom arrangement and the dimension of different molecular layers.
The graph theory gives some theorems of the fullerene regarding the established hypothesis and facts. The fullerene graphs use algebraic concepts to support the expressions. The structural configurations of the fullerenes can be given in the form of matrices or diagrams which follow a given format. The graph theory simplifies the representation and definitions by establishing given rules and definitions.
This paper gives the general background of the fullerenes and the general mathematics involved in the modeling of the fullerene structures. The graph theory explains the arithmetic in fullerene theory by use of the algebra concepts.
Chapter 1: IntroductionHistory of Fullerenes
The discovery of fullerenes and its structural elucidation paved the way for defining fullerene graphs. The discovery of fullerenes is attributed to the contribution of the Nobel laureates Harold Kroto, Robert Curl, and Richard Smalley [4; Section 1]. These scientists discovered a novel allotrope of carbon in which the carbon atoms formed close shells.

The first fullerene molecule (C60 structure consisting of 60 carbon atoms) represented a football or a dome-like structure [4; Section 1]. Moreover, this new allotrope mimicked an icosahedra structure called “Buckminsterfullerene.” Buckminsterfullerene was named after the renowned architect Buckminster Fuller who was known for designing geodesic domes during the 1960’s [1; Section 1]. However, the chemical properties of fullerenes and C60s became evident after the contributions of W. Kratschmer and D. Huffman. For the first time, Kratschmer and Huffman produced isolable amounts of C60 by their famous graphite experiments [1; Section 1]. The authors extracted the carbon condensates burning graphite rods in an atmosphere of helium. These carbon condensates were none other than fullerenes [1; Section 1]. Such discoveries formed the roadmap for unraveling the chemistry of fullerenes [1; Section 1].
Fullerene Chemistry and its Association with Graph theory
The establishment of C60 structures and the genesis of fullerene chemistry raised significant interests in fullerene graphs. Most of the research on fullerenes is centered on buckminsterfullerene [1; Section 1]. Most of the properties of fullerene graphs are based on their cyclically connected symmetries. Any graph is considered to be cyclically connected by a certain number of edges if the graph remains inseparable into two [2; Section 1]. Došlić defined a fullerene graph as a “planar, 3-regular and 3-connected graph, twelve of those faces are pentagons, and the other remaining faces are hexagons [2; Section 1]. These same structures are referred as fullerenes or trivalent chemical cages in the chemical literature [1; Proposition 1]. There is a close connection between the structural chemistry and the graph theory on the mathematical model of fullerenes [3; Section 9.8]. Any graph (say X) consists of different points which are interconnected with each other through lines. We denote the vertex set by V(X) and the edge set by E(X) [3; Section 1.1]. A simple graph is a graphical plot without any loops or parallel edges. A chemical molecule can also be represented by a simple graph whose vertices and edges represent the atoms and the chemical bonds. Hence, fullerene graphs are representative of molecules consisting entirely of carbons [3; Section 9.8].
Purpose of the Dissertation
The present article expounded on the importance and objective of the study on structure-function properties of fullerenes. The article presents a brief preview and collation of seminal works by various authors on the referred topic. The mathematical models of fullerenes and fullerene graphs can be explained by the graph theory [3; Section 9.8]. The Chemical graph theory speculates that chemical or biological molecules can be classified from the topological constituents of their structures [5, Section 1.6]. Gsaller [9] contended that such graphs could be used for modeling novel structures with new and predictable properties. Hence, the primary aim of this article is to explore the graph theory in light of the mathematical structures and properties of fullerenes and their correlation in determining the stability of the fullerene molecules. This article supports the hypothesis that there is a close connection between mathematics and structural chemistry.
Chapter 2: Background2.1. An Overview of Algebraic Graph theory
A graph is defined to be any mathematical shape or structure which connects a set of points based on a particular virtue, value, function, or rule. These points are referred as vertices, and the lines those connect these points are called edges. On the other hand, the degree of a vertex (a point on the graph) reflects the total number of edges that are connected with that specific vertex [3; Section 1].
2.2. Graph Theory in Defining Structure-Function Relationships: Experimental insights
The graph theory speculates that chemical or biological molecules could be classified from the topological constituents of their structures [3; Section 1]. Gsaller [9] contended that such graphs were beneficial in elucidating novel molecular structures. In 1985, Kroto et al. explained the abundance of C60 clusters in the atmosphere. In 1990, Kratschmer et al. confirmed the assumptions of Kroto et al. (1985) after they introduced a method for bulk production of fullerenes which was validated through IR spectroscopy [9]. Different types of theoretical tools are implemented to develop molecular models for chemical or biochemical reactions. Such modeling finds extensive use in the field of physical sciences, chemical sciences, medical sciences, and toxicology [4; Section 1].
Gaaller [9] explored the quantitative structure-property relationship (QSPR) for the physicochemical properties of fullerenes (for example, melting point) regarding their structural orientation in three-dimensional conformation. The authors highlighted that molar graphs are effective in representing the structural orientation of a chemical molecule. The structural invariants of a molar graph include its topological indices, polynomials, and spectra. Gaaller [9] described two QSPR methods to explain such graphs. In the first instance, the authors contended that the square root of any Laplacian polynomial in a molar graph describes the distribution functions of the radiation and gyration of a molecule. Secondly, the HOMO-LUMO separations and graph branching were estimated from the minimum and maximum values of the spectral features of these graphs [5]. Gaaller [5] generated an algorithm to demonstrate the structure-function correlates of 18 fullerene molecules from C20 to C80. The ball-stick models and Schiegel diagram was used to describe the molar graph of fullerenes.
Chapter 3: Molecular Topology of Fullerenes3.1. Graph theory and Molecular Topology of Fullerenes
Molecular topology refers to the mathematical description of molecules [5; Section 1]. Combining the molecular topology with the graph theory and the physical-chemical properties of molecules (such as its electron density, covalent and ionic potentials, and melting point) helps to comprehensively elucidate the molecular structure of different biological and compounds [5; Section 1]. Moreover, such depictions are useful in studying the bonding properties (the number of vertices, neighbors, and reciprocal relationships) of the referred molecules [3; Section 1]. If the features of the organic structures are merged at their turn with ordering and aromatic properties, such descriptions help to model and predict the internal structure of such molecules [3; Section 1]. It is also contended that the molecular topology coupled to the chemical graph theory helps to explain the biological activity of different toxicants and drugs on their target receptors [1; Section 1].
All fullerenes are cyclically connected at its four edges [2; Theorem 1]. However, cycles of three and four are not sufficient to explain the four-edged connectivity of fullerenes. For example, 3-connected and 3-regular planar graphs having a girth of five are not cyclically connected at the four edges. Such observation helps to redefine the structure of fullerene or fullerene graphs. Rather, the icosahedron symmetry could explain the edge-connectivity of a fullerene graph. Fullerenes exhibit twenty hexagonal and twelve pentagonal rings for its icosahedron symmetry close caged structure [2; Theorem 1]. The electron bonding and geodesic factors in fullerenes account for their structural stability [1; section 1]. Theoretically, it signifies that an infinite number of fullerenes and their respective graphs can exist based on the orientation of the hexagonal and pentagonal rings as permissible for generating icosahedrons symmetries. The smallest of these fullerenes is a regular dodecahedron that contains no hexagons at all. It is followed by the next smallest fullerene with two hexagons separated by twelve pentagons. Despite the fact that there are various structures which contain twelve pentagons, they can be represented by Euler’s formula. To recall, Euler stated that if a planar graph has n vertices, e edges, and f faces, then  n-e +f=2 [3; Section 1.8].
Chapter 4: Symmetry of Fullerenes4.1. Algebraic Graph Theory in Explaining Stability and Symmetry of Fullerenes
Fullerenes are graphs where the vertices represent the carbon atoms, and the edges represent the bonds between them [1; Section 1.1]. It was also established that fullerene graphs are 3-connected, and 3-regular planar graphs have pentagonal and hexagonal faces [1; Section 1.1]. It was further noted that fullerene graphs were possible for all C20 and >C24 fullerenes and no fullerene had two adjacent pentagonal faces. The meaning that five hexagons surrounded each pentagonal face. That is referred to as the isolated pentagon rule for fullerenes. Grunbaum and Motzkin showed that if the isolated pentagon rule is followed and the pentagonal faces are equally distributed, the resultant fullerenes are not only stable but icosahedrally symmetrical [1; Section 1.1]. To recall, the smallest unit of such symmetrical icosahedrons is a dodecahedron. However, in this section, the higher structures of symmetrical fullerenes are explained.
Andova et al.[1; Section 1.1] reported that icosahedra symmetry is based on the structural correlates of geodesic domes. Geodesic topography signifies triangulation in spheres having vertices with degree five and six [1; Section 1.1]. Thus, all icosahedral fullerene graphs are obtained by mapping the hexagonal grid over the triangular faces of the icosahedrons [1; Section 1.1]. The authors also showed that the number of vertices (n) in the icosahedron is determined by two integers i and j [1; Section 1.1]. These integers represent two dimensional Goldberg vectors or Coxeter coordinates. The relation between these two vectors and the number of vertices (n) is represented by the Goldberg equation as [1; Section 1.1]:
n = 20 i2+ij +j2Since the pentagonal faces in fullerenes are not considered adjacent, therefore; mirror effects are not possible [1; Section 1.1]. Under such circumstances, it can be assumed that I should be greater than 0 but less than j, while j should always be greater than 0 [1; Section 1.1]. This resultant vector would define the distance and positions of the vertex of I and j triangle in the hexagonal lattices [1; Section 1.1]. Therefore, 20 such triangles would form icosahedron fullerenes, and the vertices of these triangles can be considered as the centers of the twelve pentagons [1; Section 1.1]. A pair of such triangles would represent opposite triangles, and its pentagonal counterparts are referred as antipodal pentagons [1; Section 1.1]. The icosahedral group represents all possible rotational symmetries in an icosahedron or a dodecahedron with an order of 60. Therefore, all icosahedron fullerene graphs with I> 0 would exhibit icosahedral symmetry [1; Section 1.1]
Chapter 5: Algebraic Graph Theory5.1. Structural-functional correlates of fullerenes in Light of the Algebraic Graph Theory
Although it is difficult to observe fullerenes in nature, their structures are well-elucidated. Graph theory provides a further understanding of the structural-functional correlates of fullerenes. The planar representation of any chemical molecule is referred to as the Lewis dot structure. Thus each atom in a molecule can be represented by chemical symbols and lines if bonds connect them. Lewis dot structures can be easily extrapolated to a graphical representation if each atom in a given molecule considered as vertices –sharing edges, provided the atoms in it are bonded to each other. In a Buckminsterfullerene (C60), each vertex remains connected to three other vertices. Since fullerene graphs have pentagonal and hexagonal faces only, they can be simply considered planar projections of Platonic and Archimedean solids [2; Section 3].
Planar projections or planar graphs are referred to as those which can be drawn or presented in a plane with no crossing between its edges. Leonard Euler contended that for any planar graph the summation of its vertices, edges, and faces should alphanumerically equal 2 {v- e+ f =2}. Applying this theorem to the fullerene molecule, it can be concluded that fullerene has p pentagons and h hexagons. Since the faces of a fullerene graph are only hexagons and pentagons, there are five vertices per pentagon and six vertices per hexagon. Considering a fullerene consists of p pentagonal faces and h hexagonal faces, the number of vertices should be equal to 5p +6h. However, applying Euler’s rule to fullerene chemistry can be misleading. This is because the formula v=5p + 6h does not hold true if the fullerene graph lies in more than one face [3, Section 1.8].
Since a fullerene graph is considered to be 3-regular, each vertex can be considered to comprise of three faces. Hence, each vertex of a fullerene graph will be shared by three faces as v=5p +6h/3  [2; Section 3]. Likewise, a similar argument can be developed for the number of e in a fullerene graph. As per the Euler’s theorem, it is contended that the total edges in a fullerene graph (e) would be equal to 5p + 6h [3; Section 1.8]. Since the edges of the graph separate the faces, each edge can be contended to have two faces which put the total number of edges as 5p +6h/2. The total number of faces in a fullerene molecule is the summation of pentagons, and hexagons in it(f= p +h). Applying Euler’s theorem on pentagons, the assumptions can be presented as [3, Section 1.8]:
5p +6h3 -5p +6h2+ p+h = 2……………………………….Equation 1
When implementing basic arithmetic of rearrangement, Equation 1 can be expressed as
10P/6 +2h -15p/6 -3h +6P/6 + h = 2……….Equation 2
Combining different terms and canceling h in Equation 2 we derive Equation 3 as [3, Section 1.8]::
p/6 = 2………………………………………………Equation 3
Hence, it can be proved that a fullerene graph would have at least 12 pentagonal faces. Such assertions are extremely important because theoretically, one can assume a fullerene graph to have trillion vertices [3, Section 1.8]:. Although the representation of such structure is not feasible graphically, such deductions can elucidate the structural conformation of a fullerene molecule irrespective of the number of atoms in such molecule [3, Section 1.8]. The deduction of 12 pentagonal faces in a fullerene molecule is also beneficial in defining other characteristics. Replacing p=12 in basic Euler’s formula, the total number of vertices (v) comes to [3, Section 1.8]::
v= 5* 12 +6h ……………………………………………….Equation 4
It is important to consider the basic structure of smallest fullerene to have only pentagonal faces, the number of hexagons (h) should be considered 0 in Equation 4. Replacing h= 0 in Equation 4, it reflects that there are 60 vertices in the smallest fullerene [3, Section 1.8]:. However, IR spectroscopy contends that there are only 20 vertices in the smallest fullerene. Such observations are justified if the assumptions are replaced by the modified Euler’s formula for fullerenes. The modified Euler’s formula for fullerenes graph contends each vertex is comprised of three faces [3, Section 1.8]:. Replacing p=12 in modified Euler’s formula for fullerenes, the number of vertices in a fullerene graph comes to [3, Section 1.8]::
v= (5* 12 +6h)/3………………………………………………….Equation 5
Or, v= 20 +2h ……………………………… ……………………..Equation 6
However, if h is considered to be 0 in Equation 6, then the number of v (vertices) is equal to 20 (the number of vertices that should be present in the smallest fullerene as confirmed by IR spectroscopic data) [3, Section 1.8]. This basic graph of the smallest fullerene is dodecahedral because it represents the planar projections of a dodecahedron] (Doˇsli´c, 2013). This dodecahedral graph typically conforms to the molecular structure of the C20 fullerene. Choosing higher values for h would generate different fullerene graphs [3, Section 1.8].
Limitations on the Structural feasibility of 22C Fullerenes
A C22 fullerene is not possible as per the assumptions of the graph theory. If h is considered one (h=1) for equations 5 and 6, the number of vertices or carbons will be equal to 22. Theoretically, it describes the formation of a 22C fullerene. Such fullerenes would comprise of 12 pentagonal faces and one hexagonal face. Putting the above assumptions in Euler’s model (10P/6 +2h -15p/6 -3h +6P/6 + h = 2) we get [5; Section 2]:
22 – 5*12 +6/2 +12+1=2………………………….Equation 7
Or, 22-33+13= -11+13= 2………………….…Equation 8
Equations 7 and 8 reflect that Euler’s formula should hold true for 22C fullerenes. However, construction of this fullerene graph requires insertion of the hexagon into the middle of the dodecahedral symmetry of a simple fullerene [5; Section 2]. On the contrary, insertion of the hexagon in the center of a dodecahedral structure will force the outer face of the dodecahedron to shape up as a hexagon for preserving its 3-regularity. Grunbaum and Motzkin (1963) provided conclusive evidence regarding the non-existence of 22 fullerenes [5; Section 2]. These authors also proved that fullerenes with carbon atoms more than 60 (C62, C64, C66, and C68) could not prevail in nature [5; Section 2]. These fullerenes would require 21, 22, 23, and 24 numbers of hexagons for getting inserted in the dodecahedron domain. Apart from these exceptions, a fullerene with an even value of vertices that is either equal or greater than 20 is always feasible [5; Section 2].
Chapter 6: Theories, Propositions, and Proof of Structural properties of Fullerenes6.1. Perfect Matching
Definition 6.1 [5, Section 3.5] A matching M in a graph X is a set of edges of X where no two edges have a vertex in common. The number of edges in a matching is called the size of a matching.
Definition 6.2. [5, Section 3.5] Any vertex in a graph X that is incident with an edge in a matching M is said to be covered by M. A matching is called a perfect matching if it covers every vertex of X. In the chemical sciences, perfect matchings are referred to as Kekul´e structures [2, Section 3].
Theorem 6.3. [2, Section 3] Every fullerene has a perfect matching.
Each graph is considered to exhibit perfect matching (M): The term “matching” for any graph (say X) refers to E (X) {the set of edges} of X, where no two edges would have their vertices in common [2; Section 3]. The number of edges for a matching denotes the magnitude M. Hence, any vertex from a vertex set  {where v is a component of  V(X)} that is incident with an edge (e) in edge set   {where e is a component of   E(X)} is covered by M. A matching is considered to be absolutely perfect if it covers every vertex of X. In chemical sciences, perfect matching is referred as Kekule structures [2; Section 3].
All fullerene graphs are 2-extendable.: Any graph is considered to be n-extendable for the clause: 0 < n < p/2 if it is close connected (4-connected in fullerenes), bears a matching of size n, and such matching has the capability to exhibit perfect matching [2; Section 3]. Graphs which are 0-extendable refer to simple connected graphs having a perfect matching. On the contrary, 1-extendable graphs contain certain lower bounds regarding the number of the perfect matchings [2; Section 3]. It is to be noted that n extendibility implicates n-1 extendibility. Extrapolating such theorem if X is a cubic and a 3-connected planar graph with cyclically four-edge connected having no faces of size four, then X can be assumed to be 2-extendable [2; Section 3]. On the contrary, each fullerene graph can also be considered to be 1-extendable because 1-extendible graphs having p number of vertices and q number of edges would contain at least   1-p/2 =2 number of M. Hence, each fullerene graph with p number of vertices bears at   p/4 + 2 number of M [2; Section 3]. Considering that each fullerene graph is cubic with  q= 3p/2 the claim is aptly supported. Moreover, with such assumptions, the lower bounds of a perfect matching in X are also feasible [2; Section 3].
Theorem 6.4. [2, Section 3] All fullerene graphs are bicritical
A bicritical graph is one in which Xu-v exhibits a perfect match for every pair of its distinct vertices (u) [2; Section 3]. A bicritical graph that is 3-connected is referred to as a brick. Any bicritical graph X with p vertices will have p/2 +1 number of perfect matchings. If the graph X is also 2-extendable (having a minimum of 6 vertices), then it can be referred as either a bicritical graph or an elementary bipartite [2; Section 3]. If every edge of any graph displays a perfect matching, such graphs are referred as bipartite [2; Section 3]. Since all fullerenes are 3-connected, the bicritical graph of fullerenes can be referred as a brick [2; Section 3].
Theorem 6.4. [2, Section 3] All fullerene graphs are bricks
Earlier explanations indicate that any fullerene will have at least 20 vertices [2; Section 3]. On the other hand, fullerene graph cannot be bipartite because there are lower bounds to perfect matching. Considering that even the smallest fullerene will be 2-extendable, will not be bipartite, and will contain at least six vertices, it can be assumed that all fullerene graphs are bricks [2; Section 3].
Theorem 6.5. [2; Theorem 1] Any fullerene graph is always cyclically 4-connected. Considering a fullerene graph of X with a cut-set C (having vertices v1, v2, v3), so that both the components X’ and X’’ of X-C contains a cycle [2; Section 3]. Theoretically, nine edges could emanate from the cut-set towards rest of the fullerene graph. On the other hand, at least three such edges would connect C and X’ [2; Section 3]. However, if the number of edges is exactly three, there is a hindrance in forming the fullerene graph [5, Section 1]. It is because X is considered to be cyclically 4-edge connected. Likewise, no component of X-C could be connected to the cut-set by more than five edges [2; Section 3].
Considering that only four edges are possible from C to X‘ and five edges to some other component, then two vertices of the cut set (say v1 and v2) would issue one edge towards X’ [2; Section 3]. On the contrary, the third vertex of the cut set (v3) would contribute two edges toward X’ [2; Section 3]. However, the edges between v1 and v2 and X’ and the edge between v3 and x’’ will form a set of three edges in total. However, the removal of them will form two components with one cycle [5]. Such formations can be rejected because any fullerene molecule has to the cyclically 4-edge connected. Therefore, Hamilton cycle is considered to be present in all fullerenes which have < 42 number of vertices [2; Section 3]. On the other hand, 4-connected cubic planar graphs with 40 vertices are considered to be Hamiltonian. Hence, every fullerene graph should be considered a Hamiltonian [2; Section 3].
Theorem 6.5. [2; Section 3] Any fullerene graph with 176 vertices is Hamiltonian:
Hamilton cycles are existent on planar graphs and are equivalent to a three-partition. A tree-partition for a graph X is considered a partition (P1, P2) of P(X) so that both the graphs that are induced by the respective partitions are trees. The tree-partitions are balanced if the modulus of both the partitions is balanced (Doˇsli´c, 2002). Hence, tree-partitions are not guaranteed in all planar graphs. However, because the tree partitions are balanced in fullerene graphs, they exhibit tree-partition [2; Section 3].
Theorem 6.6. [2; Section 3] All Fullerene Graphs are planar and are embedded in a sphere:
According to the Euler’s formula, let X= (V, E) be a planar graph with v number of vertices and e number edges. Considering there is an embedding in graph X with f faces, Euler’s formula is given by v- e+ f= 2 [3; Section 1.8]. Strong induction on the edges can prove this embedding. For e= 0 fig A prevails, while for e=1 fig b will prevail [3; Section 1.8].
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Fig a
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Fig B
Assuming k € N for connected planar graphs with e edges (the number of edges should be >0 but < k). Hence, X can be considered a planar graph with k+1 edges and v vertices with an embedding to give f faces [3; Section 1.8]. If an edge is deleted from X, it will form a subgraph X1= X- (a, b), where a, b signifies the edge [3; Section 1.8]. Under such circumstances, X1 can be either connected or disconnected. If X’ is considered to be connected, then neither of its vertices (a, b) could have a degree of one in X. As a result, the edge (a, b) would act as a boundary between the two faces in the embedding of X and will merge in X’. Considering such assumptions, x’ would have v vertices, e-1 edges, and f-1 edges. the Euler’s theorem is obeyed when substituting these values in the induction hypothesis.
2= v – (e-1) + (f-1) = v- e+ f………………………….Equation 9
On the other hand, if X’ is considered to be disconnected then X’ will have two components (X1’ and X2’) with vertices, edges, and faces as v1, e1, f1 and v2, e2, f2 respectively. However, as per convention v1 +v2 =v and e1 + e2 = k= e-1, while f1+f2= f =1 because the infinite face should be counted twice. Applying the inductive hypothesis for both X1’ and X2’ we get [3; Section 1.8]:
v1 –e1+ f1=2 and   v2 –e2+ f2=2 Or, 4 = v1 –e1+ f1 + v2– e2+ f2
Or, 4 = (v1 +v2) – (e1+ e2) + (f1+ f2)
Or, 4 = v – (e-1) + (f +1) = v- e + f + 2 The above assumption justifies that the graph only can not only prove embedding and planarity of fullerene graphs, but it also shows that sub-graphs of fullerenes can be either connected or disconnected[3; Section 1.8]. Fullerene graph is a 3-regular planar graph in which each of its faces exhibits an embedding of either degree five or six [3; Section 1.8]. Therefore, fullerenes should have exactly 12 faces of degree 5 [3; Section 1.8]. Considering a fullerene graph X has v number of vertices, e number of edges, f5 number of faces for degree 5 and f6 number of faces for degree 6. Hence, the total number of faces (f) in X = f5 + f6. According to the Euler’s formula [8, Section 2],
f = 2+ e –v                          Or, f5 + f6 = 2+ e –v                          Or,  f5 + f6 = 2+ 3n/2 –v = v- e + f + 2
Or, f5 + f6 = 2 + v2 As per standard handshaking, the three-regularity of X will produce 2e = 3v. On the other hand, handshaking of the planar graphs will yield 2e = 5f5 + 6f6 [8, Section 2]
Hence, three v = 5f5 + 6f6     Or 3v = 5 f5 + f6 + f6     Or, 3v = 5 (2 + v/2) + f6Such assumptions can be replaced with v= 2f6 + 20 as denoted earlier, where f = hexagon face f6
Hence, f5= 2 + v2 – f6 However, f6 = v2 – 10  Hence, f5= 2 + v2 – v2 -10                                                   Or, f5 = 12……..Proposition 1
Such findings prove that any fullerene will have exactly 12 faces with a degree 5 [5, Section 2].
Theorem 6.7. [2; Section 3] No fullerene can have two adjacent pentagonal faces:
These findings further complement the notion that no fullerene can have two adjacent pentagonal faces. It is because the pentagonal orientation of the carbon atoms is not a favorable conformation for any allotrope of carbon. The stress of bearing carbon atoms in two such adjacent faces would make the fullerene molecule unstable. Hence, a fullerene molecule should have all its pentagonal faces isolated from each other. As per proposition 1, a fullerene molecule must contain 60 vertices if it has precisely 12 faces. Such proposition is has translated into the viability of buckminsterfullerene [4].
Theorem 6.8. [3; Section 2]. Adjacency matrices of fullerene graphs should have as many positive Eigenvalues as per their negative Eigenvalues:
The graph theory also contends that the adjacency matrices of fullerene graphs should have as many positive Eigenvalues as per their negative Eigenvalues. Such balancing is necessary for the stability of the fullerene molecule. It can be contended that too many Eigenvectors in either direction could destabilize the fullerene molecule. The leapfrog graph of a 3-regular planar graph X with v number of vertices, e number of edges, and f-number of faces could elucidate this corollary. First of all, the graph X is converted to a line graph L (X).
The line graph can be formed by taking the edge set of X as its vertices and connecting the two edges (with the consideration that the two edges are incident in X)IIIt turns X into a popular 4-regular planar graph having m number of vertices and n=f number of edges [1; Section 2]. Now if the graph X is split along each vertex of L(X) into a pair of adjacent vertices [8; Section 2]. The adjacent matrices are obtained in such a manner that each triangular face of L(X) represents a hexagon. Under such assumptions, the result is a 3-regular planar graph with the 2m number of vertices, length 6 for n number of faces, and an equal number of non-hexagonal faces as present in the original graph [4; Section 2]. Theoretically, it suggests that any fullerene can be converted into another fullerene. Therefore, it can be contended that if X is a fullerene graph, then the line graph of X will also be a fullerene with an equal number of positive and negative Eigenvalues.
Theorem 6.8. [3; Section 3]. Fullerene graphs do not exhibit Konig property:
Since every fullerene, the graph is a brick, and being a brick there is even subdivision of K4 (graph on four vertices) and C6 (cycle on six vertices) in its ear decomposition [2; Section 3]. However, it is contended that Konig property can only exist for graph if it does not have a nice subgraph which represents even distribution of K4 and C6 [2; Section 3]. Although K2 (graph on two vertices) and path on four vertices are nice subgraphs of a fullerene graph, K4 and C6 are not nice subgraphs of any fullerene graph. Hence, all fullerenes fail to exhibit Konig property [2; Section 3].
Theorem 6.8. [3; Section 3]. The number of perfect matchings in any fullerene graph is >3/8v and < than v/2-2 (where v represents the number of vertices):
This is because the independence number of any n-extendable graph cannot exceed v/2 –n (as per the principle of right inequality) [2; Section 3]. The principle of left inequality also holds true for triangle-free planar graphs (such as fullerenes) [2; Section 3]. Although the upper bound for perfect matching cannot be improved, the lower bounds of the same can be achieved for certain fullerenes (fullerenes bearing isolated pentagons) [2; Section 3].
Theorem 6.8. [3; Section 3]. The limits of saturation number in a fullerene graph are greater than v/4 +1 and lesser than v/2 -2 (where v represents the number of vertices):
Saturation number reflects the minimum number of maximal matching in graph X. The perfect matching of X-u-v in the bicritical graph represents maximum matching [2; Section 3]. The lower limits of the saturation number are limited from the 2-extendable property of fullerenes (because the saturation number in n-extendable graphs should be at least v/4 +v/2) [2; Section 3].
Chapter 7: Predicting Molecular Structures of Tetrahedral Fullerenes7.1. Graph Theory in Predicting Molecular Structures of Tetrahedral Fullerenes
The chemical graph theory can often be used to predict the molecular structure of mini-fullerenes [5; Section 2]. A tetrahedron can be denoted by any of the two structures (Fig A or Fig B). Fig A is considered to be the molecular structure of a 4C fullerene if a reaction-active single atom initiates the formation of the fullerene [5; Section 2]. On the contrary, Fig B proposes that the graph is initiated only as a dimer of two carbon atoms. However, both these graphs have four vertices and six edges and are isomorphic with each other. The graph theory states that edges that have a common vertex are known as an adjacent [5; Section 2]. A graph that has a one-to-one correspondence, and can conserve the adjacency, is indeed isomorphic (Doˇsli´c, 2008). On the other hand, both graphs can be considered planar because they are obtained on one plane only [5; Section 2]. However, the graph (Fig A) resembles a regular tetrahedron more than the abstract graph (Fig B).
The assumptions are based on visual observation. For example, when one looks Fig A from above, it resembles a three- dimensional triangular pyramid. On the contrary, Fig B can be observed with normal to skewed edges [5; Section 2]. The graph theory contends that no two edges in a fullerene graph should intersect each other, and the graph itself should be planar. These assumptions are met more by Fig A compared to fig B. Hence, 4C tetrahedral fullerenes can be better expressed as Fig A compared to Fig B. Moreover, it is also contended that a planar graph is more suitable to capture the symmetry of a corresponding polyhedron. As a result, planar graphs help to appropriately elucidate the structural details of a polyhedron [5; Section 2].

Fig A: Tetrahedron Fullerene [5, Section 2]

Fig B: Abstract representation [5, Section 2]
5.3. Innovations in Graph Theory: Molecular Structure of Truncated Elementary Fullerenes
A truncated tetrahedron can explain some of the innovations in the graph theory. In a tetrahedron that is truncated, a reaction-active cluster of three atoms is evident [5; Section 5a]. This reaction-active cluster initiates the genesis of a truncated tetrahedron [5; Section 5a]. In the perspective of the graph theory, each atom in a truncated tetrahedron could be considered a vertex [5; Section 5a]. On the contrary, if the reaction-active clusters are considered a big vertex, then one can achieve a fullerene structure of a regular tetrahedron. Such deductions from the graph theory help to analyze the molecular structures of truncated tetrahedrons [5; Section 5a].
Chapter 8: Report on Advanced Explorations8.1: Fractal Geometry in Fullerene Crystallography
Fractal geometry shapes have the property of self-similarity, the assets that a shape haves its pattern repeated on different scales. The fractal definition aims to explore fractal natures behind fractal sets and fractals in real life. I am deeply amazed by fractal properties. It is truly one of the shining gems in the crown of mathematics.
Points can form fullerene crystal as a result of taking each point and applying transformations Xa, Xb, and Xc to the points randomly. A set of points labeled as A, B, and C, forms the vertices of the fullerene crystal at any random vertex X1. Another set of points and random point X2 forms a subset of the fullerene crystal. The random sets of the points will form fullerene crystal if the sets of the points form the following equation.
Xn+1=12(Xn+Am)Where n is a random number. If the first points lie within the perimeter of the formed triangle, t hen none of the other vertices lies within the triangle. If the vertex X lies outside the triangle, the triangle will be infinite.
The triangle is constructed using the following steps:
Take 3 points in the plane which forms a triangle
Randomly choose any point inside the triangle while considering the current position
•Choose any other three vertices
•The points are translated half distance
•The current position is plotted
•The above procedure is repeated for the rest of the points
The Fullerene crystal is similar to the Graphical theory correlation in fractal geometry. The fullerene crystal forms a pattern that conforms to an iterated function system. The point outside or inside the Graphical theory correlation with very points of the leftover points. On the other hand, if the starting points lie outside the fullerene crystal, the other vertices lie inside the triangle. The outline of the triangle is formed after a hundred points are plotted.
The fullerene crystal provides a method of constructing or creating a fractal geometry. The initial point is selected at any random point of the polygon. The iterative sequence of the points gives the fractal curve of the polygon. The vertex is selected at random during each iteration. Each point lying on the curve gives a fraction of the distance between the previous points and the successive points (Godsil, 2001). If the iteration is repeated for a large number of times and the selection of points is made in each iteration, then the polygon is the fractal shape. If the initial points are four, then the resulting polygon is a fullerene tetrahedron polygon. If the number of the initial points increases to infinity, the corresponding polygon is called the Sierpinski’s simplex.
The creation of the fullerene crystal involves the concept of the iterated function system. The iterations must converge to the particular point of the iterated function system. If the initial points belong to the attractor of the iterated function system, then it the resulting polygon is a dense set of the triangle. The fullerene crystal plot point in the random is order in the attractors involved [1, Section 2.0]. This method is so different to the methods of the drawing fractals. The shape of the fractal shape is plotted easily concerning the chaos game. It is difficult to plot the polygon concerning the details of the fractal shape involved.
In the fullerene crystal is run with a square, then the fractal polygon appears, and the squares fill the shape fills evenly in the points. The restrictions of the distribution of the points or the vertices determine the shape of the fractal shapes formed. Fractal geometry exhibits the property of the self-similarity. The Graphical theory correlation shows the three-way recursive algorithm. The procedure of producing the fullerene crystal by a free hand is very simple.
left35179000
Figure C: Fullerene Crystal Pattern
The fullerene triangle has rows which form an approximation of the triangle pattern. The fractal triangle consists of the pattern built by the binomial coefficients. The rows of the fractal triangle conventionally enumerated with the initial value of the row are zero. The initials of the rows are numbered from the left of the triangle. The triangle is constructed with a formula. The rows increase from the initial row down to the final row. In mathematics, the formulae for constructing the fractal triangle is well defined with the fractal shape.
The fractal shape is defined by any nth and the column of the fractal triangle. The recurrence of the coefficients is set of the entry of the rows. Fractal triangle has high dimensional generalizations. The shape forms the new version of fullerene pyramid. In obtaining the pyramid of the fractal triangle, the coefficients of the pattern form fractal known as fullerenes triangle. The shape of the fullerenes triangle results in pattern assuming fixed perimeter [1, Section 2.1].
The fullerenes triangle has some distinct paths and square. The number of the fullerenes triangle coefficients forms a pattern that corresponds to fractal geometry. In the fullerenes triangle, the distribution of the coefficients of the probabilities. The matrix exponential gives the construction of the factorial of the coefficient. The fullerenes triangle forms exponential function with matrices with a sequence of 1, 2, 3 and 4. It has sub-diagonal, and it has zero value everywhere.
The Graphical theory correlations generated using the fullerenes triangle. The number of the iterations determines the fractal relationship generated. The fractal geometries can be generated by eliminating the triangles that are upside-down from the original pattern and then forming iterations [4, Section 1.8]. The fractal geometry is generated by duplicating the small triangles and re-arranging them into fractal geometry using iterative steps.
Fullerenes polygon movement of the molecules between the three points. The limit of the molecules goes to the infinity, and the pattern of the resulting graphs is interpreted as a Graphical theory correlation.
8.2 The Fullerenes polygon.
The fractal geometry has the property referred to as Hausdorff dimension expressed;
log3.0log2.0≈ 1.585The concept shows that half scales each iteration. If the fullerenes polygon is taken as the base of generating the fractals triangle, the result is an approximation of the rows. The area of the fractal’s triangle is zero. The concept is proven by the infinite iterations where only a quarter of the area is left at the previous iterations. Fractal’s triangle can also be drawn using the computer program. The java programing language is used to draw the fractal geometry. The program for generating the Graphical theory correlation is object-oriented and uses the class n- fullerenes triangles.
The perimeter and the area of the fullerenes polygon are determined as follows: Consider a fullerenes polygon used in the investigation whereby the triangle is removed step by step. The number of the triangles removed at each step can be established using the geometric series pattern [4, Section 2.8]. The number of the triangles after successive iterations will be depleted as the number removable from the triangle increases exponentially. At any Kth iteration the number of the triangles in the fullerene graph theory removed is equal to the;
n=(3k -1)If the side of the triangle has unit length, then the area and the perimeter of the triangles is calculated. The area of the triangle is established with side length. The area of the equilateral triangle is determined from the formula; the graph theory defines the area of the fullerene structure as shown below.
A=x234The total area of the triangle is taken as defined by the graph theory:
A=1-( 34)kAs k approaches unity the value of the term (34) approaches unity. Thus the area of the triangle becomes zero. Each of the small triangles in the fullerenes polygon has the three sides equal. The perimeter of the triangle thus estimated form the sum of the sides of the removed triangles.
The perimeter of the graph generated is as shown below;
P=932k-1When the value of k approaches infinity the, the perimeter approaches the infinity.
The complex geometry of the fullerene molecular structure is called the Mandelbrot set. Named after Mandelbrot himself, Mandelbrot Set is the set of complex number c of an iterated function with initial condition s = 0. The definition of Z and C are as follows.
Zn+1=zn2+cc∈M ⟺limn→∞sup⁡|sn+1|≤2The fullerene structure repeats its curvy pattern on different scales: when zoomed in, there will always be a new segment of length that requires a greater scale to measure, to which if repeat this zoom-in process infinite times, the length will eventually be infinity. Later this problem becomes the well-known “fullerene paradox.” The ideology behind this notion is the study of fractal geometry, a relatively new field of mathematics. [4, Section 2.5] To figure out the truth behind fullerene word, I decide to study further details on fractal geometry. Fractal geometry shapes have the property of self-similarity, the assets that a shape haves its pattern repeated on different scales. The paper aims to explore fractal natures behind fractal sets and fractals in real life. I am deeply amazed by fractal properties. It is truly one of the shining gems in the crown of mathematics.
In this section of the exploration, I will emphasize on the algebraic definition of Mandelbrot set and proof on its property. First of all, let’s take a look at the graph of Mandelbrot set. The graph of Mandelbrot Set is one of the most famous fractal shapes in the world. The self-similarity property can be spotted when the part of the perimeter is zoomed. With some help, I managed to get some simple lines of python code that can help me generates the graph with Mandelbrot set. Such code allows me to modify the number of iterations to enhance my understanding to Mandelbrot and help my exploration on Mandelbrot set.
0351790
[Figure D: Mandelbrot Set]
The graph above is a Mandelbrot set with 10000 iterations. From left to right, the iteration times are respectively 5, 20, and 100. All of the graphs display the numbers within the current number of iterations. The number that is in the set is colored blue, deep blue and elements that escape (with distance to origin greater than 2) are colored from white to blue to which the cooler the color, the faster the escape speed is. With the number of iteration bigger and bigger, the more accurate and clear the boundary of the set is. That element in the top left and bottom left corner escapes as soon as the first iteration ends. Why is this? Take the number on the top left corner, -2+1.25i as an example.
n Z C Z2+C0 0 Initial Condition 0
1 0 -2+ 1.5i -2+1.25i
Merely with one iteration,
z1=4+2.25>2The distance of Z1 already fell out of the bound, and by definition, it is not an element in the set M. The definition has that to be in the set, the function can never go beyond 2. However, why is this?
By reading the chapter on Mandelbrot in the book Complex Dynamics by Carleson and Gamelin (p.338), I think I have understood why, mathematically, once an element makes the absolute value of function goes beyond 2, the function ends up approaching to infinity eventually.
When I am proving such property (If function’s absolute value ever goes above 2, it diverges to infinity), I found that for the complex number | c | > 2, c already makes the function’s absolute value go beyond 2 for the first iteration. I wonder If this means the graph of the first iteration of Mandelbrot set is a circle with a radius of 2? I tried to verify my guess by searching answer on the internet and here is what I got; the graph of Mandelbrot with one iteration is exactly a circle on the complex plane with a radius of 2.
However, as the iteration goes on, for the complex number | c | < = 2, the boundary becomes more and more complicated. I believe to discover and explore more truth of this; I need to dig deeper into the field of complex analysis which is far beyond my current level [4, Section 2.4].
8.3 Computing Fractal Dimension of Fullerene Molecular Structures
Discovering fractal geometry also explains the fullerenes crystal structure. Start with the simple one; fullerenes polygon is a mathematical curve bought up by Sweden mathematician Koch. For fullerenes polygon, we need to start with an equilateral triangle with each side as length = 1. During each stage the division on each side into three parts and then add an equilateral triangle based on the middle part of each segment then take the base out. We use n to denote different stages. In this sense, for base case n = 0.
For n = 0, Koch snowflake is just an equilateral triangle.
The perimeter is;
3 × 1 = 3 unitsThe area of fullerenes polygon is easily determined as shown
34×12 =34unitsFor n = 1
The perimeter is;
12×13=4 unitsThe area then becomes;
48×19=163 unitsFor n = 2
The perimeter is;
=64×127=21027 unitsThe area is thereby;
=641×227=12827 unitsIn this exploration, we only calculated first three stages of fullerenes polygon. As n goes on bigger and bigger, we can formulate the perimeter and area by analyzing its pattern. From the calculation, we got that P(n) eventually becomes infinity! Interpret this result mathematically, this means as n goes to infinity, the perimeter goes to infinity then area is still a finite number. An infinite curve encircles a finite area! As a fractal shape, fullerenes polygon also exhibits the self-similar fractal property [4, Section 1.8]. Zoom on any edge of fullerenes polygon, the shape repeats itself.
The fractal geometry involves the shapes with self-similarity. The examples of the fractal geometry include the fullerenes polygon. The Graphical theory correlation is constructed from the series of iterations. Fullerenes polygon is an example of the Graphical theory correlation. The fullerenes polygon e and Mandelbrot set are examples of the Graphical theory correlation. The triangle is a type of fractal geometry.
The fullerenes polygon concept is very useful in astronomy and computer science disciplines. Astronomers assume that universe is spread uniformly in the space. The universe is termed as large-scale made of small scales that are fractal at all scales [1, Section 2.8]. The information helps to predict information on certain spaces in the universe not reachable. The image compression in computer science uses the concept of fractal geometry. Images are compressed into complex formats. On the contrary, the image is enlarged without pixelization.
Fractal geometry is technology under research. More information is yet under research because scientists and physicist think that the concepts are used in several critical areas. Fluid mechanic engineers have established that the turbulent flows are a representation of the fullerenes polygon. Use of the concepts can then study complex flows. Aldo, the telecommunication antennae have been reduced in shape and weight by use of the fractal geometry. Fullerenes polygon thus lays the foundation for other modern technologies.
8.4 Exponential growth in Fullerenes Crystal Formation
The exponential property of fullerenes polygon growth can well be described during the initial session of growth [1, Section 2.8]. The exponential process in fullerenes polygon occurs when the rate of change due to growth is proportional to the number of atoms. For instance, considering the number of given carbon atoms to be h, then;
h=h(0)ert⋯⋯⋯⋯⋯⋯(i)Where h0=the initial number of carbon atomst = time
r = intrinsic rate of increase
The above equation has been obtained from Malthusian (exponential growth) equation;
Differentiating the Malthusian;
dhdt=h(0)rert=hrTherefore;
r=dhdthThe formula makes r be the rate of incremental change by the total height of the carbon atoms. Therefore,
lnhh0=rtConsidering the population to be t, then at t+td the atoms will have doubled. Therefore,
ht+tdht=2⋯⋯⋯⋯⋯(ii)Equation (ii) compares with equation (i) such that,
hh(0)=ht+tdht=ertd=2Therefore;
td=ln2rThus, for r being measured using percentages, then the time taken to double is approximately 70 per percentage rate.
td≈70r%Hence for example;
The growth rate of fullerenes polygon structure is 42%, how long will they take before they double in number?
The solution will be given by;
td≈7042=1.667Therefore, the fullerenes polygon structure will take 1.667 units to double.
This paper has described when the rate of change is proportional to the size or number of the subject, then an exponential growth occurs. The concept is an important aspect of learning due to the vast application in which it may find a place in mathematics. This new concept may also be applied to determine the rate of growth of very small unmeasurable creatures such as white ants. The discovery may also find application in the determination of the growth rate of the fullerene materials.
The exponential function, e, may have contained several other yet to be discovered aspects that could redefine mathematics, especially abstract algebra [1, Section 2.8]. The concept goes a long way to show the unequaled importance of the exponential function.
Chapter 9: Further Areas of Study9.1: Fullerene material properties
The graphical theory aspect for the further study includes the triangulation of the sphere with vertices of degree 5 and 6. The icosahedral fullerene graphs are obtained by mapping of the hexagonal grid onto the triangular faces of the icosahedron [1, Section 2.4]. The number of the vertices of the icosahedral are calculated by use of two methods. The equation used include: m = 2i+j+2j. The unknown letters I and j are the components of the Goldberg formula in triangulation algebra. The mirror effect is used to assume that 0 ≤ i ≤ j and 0 < j.
Fullerene is a non-metallic and non-metallic material. The material is composed of metal, non-metal or metalloid. The material consists of mainly ionic and covalent bonds. The fullerenes properties lie between the metals and the non-metals. The crystal structure of the fullerenes lies between the semi-crystalline and the amorphous structure. The bonds range from ionic and covalent bonds.
In graphical theory, the icosahedral group can be defined as an algorithm for solving of numerical problems mainly involving linear equations. The coefficients of those matrices are issymmetric and non- singular. The icosahedral fullerene graphs mostly applied in implementing iterative algorithms. This is again used in the analysis of sparse systems that are large to be broken down by direct methods or methods like Cholesky decomposition and LU decomposition. Large sparse systems have equations that come up when trying to numerically solve ordinary differential equations or optimization of problems. The icosahedral fullerene graphs are used by civil engineers to solve optimization problems like the engineering problems involving energy optimization. This method was developed by mathematicians and engineers who specialized in applied mathematics and advanced algebra [1, Section 2.2]. This method of conjugate gradients was discovered by E. Stiefel and Hestenes. Currently, the Eigen value problem can be solved using the icosahedral fullerene graphs.
The icosahedral fullerene graphs give a generalization to coefficient matrices. Most of the non-linear icosahedral fullerene graphs are used to provide the minima of any non-linear equations.
For example with the following system of equations
Ax=B
Vector x is taken to be known m× m matrix. A is symmetric matrix i.e. ( AT =A).B is known. We can denote the solution of this system by using vector x∗. We can prove that two vectors e and f are conjugates if and only if eT A f=0.If A is symmetric non–zero, the left-hand side gives the inner product. Vectors are conjugate if they are orthogonal in relations to the inner product. Conjugate is a symmetric relation: if e is conjugate to f, then also f is conjugate to e.
Let’s take the conjugate vectors g and k.We can use them to approximate the solution x.We can also use the icosahedral fullerene graphs as an iterative method. With the iterative approach, we can approximate the solution of systems though it takes a long time to carry out the iterations.
Let’s assume the initial values for x∗ by x0. The values can be assumed without any loss x = 0.If we start with x then the solution from the successive iterations can be used to get the approximate value of the system of equation [1, Section 2.1]. The solution comes from the fact that any solution to the system of the equations is unique. For example in the following equation. Conjugate constraint has an algorithm which bears resemblance to orthonomization. After a long derivation, the formula gives the following expression.
PK= rk-i<kPik ArkPiT API PiFrom the above expression, the algorithm provides a direct explanation of the icosahedral fullerene graphs. The algorithm above states has to have storage of all previous directions and vectors, in addition to many matrix-vector manipulations thus they are complex to compute them manually. However, one matrix-vector manipulation is required in each iteration. The algorithm is used for solving AX =b where A is a real and positive-definite matrix-vector. The input vector is an estimate of initial solution or zero. It has a different derivation and formulation of the exact procedure.
Consider an example of the linear system Ax = b
The following two steps of the performance of the icosahedral fullerene graphs starting with an initial guess to get the approximate solution to the system.
For reference, the exact solution is as follows:
The first step is to determine vector r0 associated with x0. This vector is calculated from the expression  r0 = b – Ax0.
Since this is the first iteration, we will use the residual vector r0 as our initial search direction p0; the method of selecting p will change in further iterations. We now compute the scalar α0 using the relationship. We can determine the x using the formula
x1= x0 +α0P0= 21+73331 -8-3=[0.23560.3384]The above formula performs the first iteration.The result is an approved solution of the approximated value of the system x1.
In the next step, we can determine the next residual vector by use of the formula provided below. The formula simplifies the lengthy processes of calculating the approximated value
x1= x0 +α0P0= -8-3+73331 4113[-8-3]=[0.23560.3384]The next step includes the computation of the step scalar β0 that is used to determine the incoming search direction P1 .
β0= r1Tr1r0Tr0= 0.281 0.7492[0.2810 0.7492[-8-3][-8-3]=0.0088By use of the value of  β0, determined from the above formulas the next search direction p1 can be computed using the following formulae.
P1= r1+β0P0=0.28100.7492+0.088-8-3= 0.35110.7229The values of the quantity α1 can be computed using the value form the previous calculation. The new value can be computed by use of the following mathematical formulae.
α1= r1Tr1P1TAP1= [-0.2810.7492][-0.28100.7492][-0.35110.7229][ 4113][ -0.3511]0.7229In the final step now the value of x2 can be computed using the similar steps as the ones used to determine the value of x1.
X2= X+α0P1=0.23560.3384+0.4122-0.3511-0.7229= 0.090900.6364The solution that is got after the calculations for x2, is more accurate method for estimation of the solution than the solution for x1 and x0.Thus we can conclude that with more iterations the solution for a given system becomes more accurate [1, Section 1.2].
The icosahedral fullerene graphs can be termed as a direct method that gives the exact value of the solution after carrying out a finite number of iterations. The method has a limit of being unstable with regard to small perturbations.
The cross-sectional area of the pellets is determined using the following formulae
A=πd24For the post-sintering process, the graph theory defines the hardness of the fullerene can be determined as follows
HV=1.8544pd2The value of the load is given as the product of the pressure and the cross-sectional area.
P=σ×AThe table below shows the hardness of the pellets before and after the sintering process.
9.2:  Topological graph theoryTopological graph theory is a special pattern in algebra got by dividing a line into two parts so that the longer part divided by; the smaller part is also equal to the whole length divided by the longer part. The symbol for it is usually phi, after the 21st letter of the Greek alphabet. In equation form, it is represented as ab=(a+b)a=1.6180339887498948420…According to pi, i.e., the ratio of a circle circumference to its diameter, the digits continuous theoretically to infinity (Andova, 2012). Phi is usually given as 1.618. In the history, the number has been seen in the architecture of many ancient creations like the truncated icosahedron. In the fullerenes structure, the length of each side of the base is 756 nanometers with a height of 481 nanometers. The ratio of base to the height is approximately 1.5717 which is near to that of Golden ratio.
Leonardo Fibonacci came up with unique characteristics of the truncated icosahedron where the sequence ties directly into the fullerene atom pattern. For example, the ratio of 3 to 5 is 1.666, the ratio of 13 to 21 is 1.625, and even higher the ratio of 144 t0 233 is 1.618.
The Fibonacci numbers are applied to the proportions of a rectangle known as the Fibonacci rectangle. The rectangle is termed as one of the most visually satisfying of all fullerene atomic pattern sequence. The truncated icosahedron rectangle is also having a relationship to Fibonacci spiral, which is generated by making adjacent squares of fullerenes dimensions.

[Figure E: Fibonacci Rectangle]
A truncated icosahedron rectangle in blue with longer side a and a shorter side b when the place to adjacent to the square with sides of length a, will produce a similar golden rectangle with a length of a+b and with withe dth a. this shows the relationship;
a+ba=ab=φ. In math, two quantities are in the truncated icosahedron if their ratio is the same as the ratio of their sum to the longer of the two quantities. Algebraically it is expressed i.e. quantities a and b with a>b>0, a+ba=ab≝φ. Where phi φ represents the Truncated icosahedron and its value is; φ=1+52=1.6180339887….The properties of the Truncated icosahedron in linear algebra which include an appearance in the dimensions of its regular fullerene pentagon and in a Fibonacci rectangle that may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio, therefore, has been used to analyze the proportions of fullerene pentagon.
9.3: Analysis
Analysis of fullerene pentagon entails some methods such as calculation method, compass, and straightedge construction methods and truncated icosahedron method. Two quantities a and b will exist in truncated icosahedron ratio φ only if; a+ba=ba=φ. Method for finding the value of φ is by starting with the left fraction. This is done through simplifying the fraction and substituting in ba=1φ, a+ba=1+ab=1+1φ. Therefore;1+1φ=φ. Multiplying both sides by φ we get; φ+1=φ2 which can be rearranged to give; φ2-φ-1=0. By using the quadratic formula, two roots s of the solutions are obtained as; φ=1+52=1.6180339887…. And φ=1-52=-0.6180339887….Since phi (φ) is the ratio of positive quantities, φ is automatically positive. φ=1+52=1.6180339887….2124075255905C
0

C

9.4: Compass straight edge construction method
The truncated icosahedron may also be obtained through geometry. The geometry is done by constructions of an equilateral triangle, square and pentagon based inside a circle and perhaps in more complex three-dimensional solids such as dodecahedrons, icosahedrons, and Buckyballs.

The first step is to divide a line segment by exterior division with the sum a=1 and hence a+b=φ. The line segment AB is about twice and the semicircle with radius AS within point S IS drawn hence the intersection point D is found. Semicircle is drawn with radius AB about point B. the arising intersection point is found to be E which is same to2φ.the the the the the The perpendicular on the line segment AE from D will be established. The following parallel FS to the segment CM is the produced which is the hypotenuse of right angle triangle SDF. The hypotenuse FS is due to the cathetuses SD = 1 and DF= 2 and a length that is equal to value of5.
At last, the he circular arc is drawn with radius 5 Around the point F. Since SD=2, line MS the circular arc meets point E by 2φ which results to2φ in in in in in in in=1+5, therefore, it indicates that;
φ=1+52=1.6180339887…Truncated icosahedron
There are a great similarity and relationship between fullerene polygon structure and truncated icosahedron;
0, 1, 2, 3, 5, 8, 13, 21, 34………….
In truncated icosahedron lattice number, when taking any successive, the ratio is close to truncated icosahedron ratio. In fact, when the number becomes bigger the closer to the truncated icosahedron ratio. The arrangement is as shown in the table below;
A B BA2 3 1.5
3 5 1.666666…..
5 8 1.6
8 13 1.625
144 233 1.618055556….
233 377 1.618025751…..
Sometimes 2 and three are ignored and chose 192 and 16 as shown below;
A B BA192 16 0.08333333….
16 208 13
208 224 1.07692308….
224 432 1.92857143….
7408 11984 1.61771058….
11984 19392 1.61815754….
The table indicates the last answers from both tables the higher the number, the closer to the Fibonacci ratio; 1.61802575. And 1.6181575The inverse golden ratio and the Fibonacci ratio φ±=(1±5)/2 have a set of symmetries that preserve and make them relate. Normally, they are preserved by the fractional linear transformations x,1-x, (x-1)/x they are reciprocals, symmetric about 1/2, and symmetric about 2.
The maps form a subgroup of the molecular group PSL (2, Z) isomorphic to the symmetry group S3, {0, 1, ∞} of three standard points on the projective line and symmetries corresponds to the quotient map.
S3→S2 the subgroup C3<C2 consisting of three cycles and the identity () (0, 1, ∞) (0, ∞, 1) fixes the two numbers while the two cycles interchange these thus realizing the map.
The Fibonacci ratio is irritation of number that is approximately to 1.618. This number can be obtained using different methods such as geometry method, calculation and truncated icosahedron method. In fullerenes atomic structure, it plays a big role in estimating the atoms arrangement [1, Section 1.6].
Godsil came up with unique characteristics of the truncated icosahedron where the sequence ties directly into the Fibonacci ratio. For example, the ratio of 3 to 5 is 1.666, the ratio of 13 to 21 is 1.625, and even higher the ratio of 144 t0 233 is 1.618.
ConclusionAlgebraic graph theory is a fascinating branch of mathematics that applies the principles of algebra in solving or defining graphical problems. The science incorporates facets of basic algebra and graph theory. Algebraic graph theory is broadly divided into three main domains; implementation of linear algebra, implementation of group theory, and application of graph invariants.
Linear algebra involves the construction of different matrices for solving different aspects of the graph theory. Group theory is integrated to explain its rationale in the context of the graph theory. Graph invariants involve the study of the abstract structure of graphs based on the theoretical possibilities of linear algebra and group theory. The algebraic graph theory was put forward by the renowned Swiss mathematician Leonhard Euler in 1735. Hence, graph theory is simply the study of graphical patterns.
References[1] Andova, V.; Doˇsli´c, T.; Krnc, M.; Luˇzar, B.; Skrekovski, R. (2012). ˇ On the Diameter and Some Related Invariants of Fullerene Graphs, MATCH Commun. Math. Comput. Chem. 68, 109-130.
[2] Doˇsli´c, T., 2002. On some structural properties of fullerene graphs, Journal of Mathematical Chemistry, 31, 187-195.
[3] Doˇsli´c, T., 2008. Leapfrog Fullerenes Have Many Perfect Matchings, Journal of Mathematical Chemistry 44, 1-4.
[4] Doˇsli´c, T., 2013. The Smallest Eigenvalue of Fullerene Graphs Closing the Gap, MATCH Commun. Math. Comput. Chem. 70, 73-78.
[5] Godsil, C.; Royle, G., 2001. Algebraic Graph Theory, Graduate Texts in Mathematics 207, Springer.

Contents
TOC o “1-3” h z u Abstract PAGEREF _Toc511627292 h 2Chapter 1: Introduction PAGEREF _Toc511627293 h 2Chapter 2: Background PAGEREF _Toc511627294 h 4Chapter 3: Molecular Topology of Fullerenes PAGEREF _Toc511627295 h 5Chapter 4: Symmetry of Fullerenes PAGEREF _Toc511627296 h 6Chapter 5: Algebraic Graph Theory PAGEREF _Toc511627297 h 8Chapter 6: Theories, Propositions, and Proof of Structural properties of Fullerenes PAGEREF _Toc511627298 h 11Chapter 7: Predicting Molecular Structures of Tetrahedral Fullerenes PAGEREF _Toc511627299 h 19Chapter 8: Report on Advanced Explorations PAGEREF _Toc511627300 h 21Chapter 9: Further Areas of Study PAGEREF _Toc511627301 h 32Conclusion PAGEREF _Toc511627302 h 42References PAGEREF _Toc511627303 h 44