# Free Fullerenes Dissertation Example

STRUCTURE AND PROPERTIES OF FULLERENES

Chapter 1: Introduction

History 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. 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: Background

2.1. An Overview of Algebraic Graph theory

Algebraic graph theory is a fascinating branch of mathematics that applies the principles of algebra in solving or defining graphical problems [3; Section 1]. The science incorporates facets of basic algebra and graph theory [3; Section 1]. 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 [3; Section 1]. 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 [3; Section 1]. Hence, graph theory is simply the study of graphical patterns. 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 [9]. Gaaller [9] 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.

2.3. Graph theory and Molecular Topology of Fullerenes

Molecular topology refers to the mathematical description of molecules [6; 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 [6; 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 [6; 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].

2.4. Algebraic Graph Theory in Explaining Stability and Symmetry of Fullerenes

To recall, 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 having pentagonal and hexagonal faces only [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. This meant that each pentagonal face was surrounded by five hexagons. This is referred 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 are 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 ne always 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].

2.5. 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 as the Lewis dot structure. Thus each atom in a molecule can be represented by chemical symbols and lines if they are connected by bonds. 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 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 be alphanumerically equals 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

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

Considering 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. To recall, the modified Euler’s formula for fullerenes graph contends t 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. 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].

2.6. Theories, Propositions, and Proof of Structural properties of Fullerenes in light of Graph theory

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].

All fullerene graphs are considered to be bicritical: A bicritical graph is one in which X-u-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 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].

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].

Any fullerene graph is always cyclically 4-connected [2; Theorem 1]: 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 [7, Section 1]. This 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].

Any fullerene graph with 176 vertices is Hamiltonian: Hamilton cycles are existent on planar graphs and is equivalent to a tree-partition [2; Section 3]. 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. 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].

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]. This embedding can be proved by strong induction on the edges. For e= 0 fig A prevails, while for e=1 fig b will prevail [3; Section 1.8].

Fig a

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. Substituting these values in the induction hypothesis, the Euler’s theorem is obeyed [3; Section 1.8]:

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]. To recall, a 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, 3 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 [8, Section 2].

No fullerene can have two adjacent pentagonal faces: These findings further complement the notion that no fullerene can have two adjacent pentagonal faces. This 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].

Adjacency matrices of fullerene graphs should have as many positive Eigen values as per their negative Eigen values: The graph theory also contends that the adjacency matrices of fullerene graphs should have as many positive Eigen values as per their negative Eigen values. Such balancing is necessary for the stability of the fullerene molecule. It can be contended that too many Eigen vectors 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). This turns X into a popular 4-regular planar graph having m number of vertices and n=f number of edges [8; 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 2m number of vertices, length 6 for n number of faces, and equal number of non-hexagonal faces as present in the original graph [8; 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 Eigen values [8; Section 2].

Fullerene graphs do not exhibit Konig property: Since every fullerene 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].

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].

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].

2.7. 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. To recall, 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 is able to conserve the adjacency, is indeed isomorphic. 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]

2.8. 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 clearly 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].

References

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