Fullerene Graphs Research Articles

Closed trail distance in a biconnected graph.

Graphs describe and represent many complex structures in the field of social networks, biological, chemical, industrial and transport systems, and others. These graphs are not only connected but often also k-connected (or at least part of them). Different metrics are used to determine the distance between two nodes in the graph. In this article, we propose a novel metric that takes into account the higher degree of connectivity on the part of the graph (for example, biconnected fullerene graphs and fulleroids). Designed metric reflects the cyclical interdependencies among the nodes of the graph. Moreover, a new component model is derived, and the examples of various types of graphs are presented.

On bicriticality of (3,6)-fullerene graphs

A (3,6)-fullerene is a plane cubic graph whose faces are only triangles and hexagons. A connected graph G with at least $$2n+2$$ vertices is said to be n-extendable if it has n independent edges and every set of n independent edges extends to a perfect matching of G. A graph G is said to be bicritical if for every pair of distinct vertices u and v, $$G-u-v$$ has a perfect matching. It is known that every (3,6)-fullerene is 1-extendable, but not 2-extendable. In this short paper, we show that a (3,6)-fullerene G is bicritical if and only if G has the connectivity 3 and is not isomorphic to one graph (2,4,2). As a surprising consequence we have that a (3,6)-fullerene is bicritical if and only if each hexagonal face is resonant.

Graph operations based on using distance-based graph entropies

Let G be a connected graph. The eccentricity of vertex v is the maximum distance between v and other vertices of G. In this paper, we study a new version of graph entropy based on eccentricity of vertices of G. In continuing, we study this graph entropy for some classes of graph operations. Finally, we compute the graph entropy of two classes of fullerene graphs.

Max-min Rodeg index of bridge graphs and fullerenes

Chemical graph theory is a branch of mathematical chemistry which deals with the nontrivial applications of graph theory to solve molecular problems. A graph-based molecular descriptors commonly known as topological indices describe the structures of chemical compounds. They are used in the isomer discrimination, structure property relationship, structure activity relations. Also, the certain physicochemical properties such as boiling point, enthalpy of vaporization, stability, and so on are able to be predicted by this technique. Molecules and molecular compounds are often modeled by molecular graph which is a representation of the structural formula of a chemical compound in terms of graph theory. Recently, the Max-min rodeg index (Mm sde ) which is vertex degree-based topological index has attracted attention and gained popularity. This index give the best predictor for enthalpy of vaporization and standard enthalpy of vaporization in the set of octane isomers and also for log water activity coefficient in the set of polychlorobiphenyles. A fullerene graph is a cubic planar graph whose faces are pentagons and hexagons. In this study, the Max-min rodeg index of fullerenes and link of fullerenes is computed. Moreover, it is presented exact expressions for the Max-min rodeg index of bridge graphs.

On partition dimension of fullerene graphs

Let $G = (V(G), E(G))$ be a connected graph and $\Pi = \{S_{1}, S_2, \dots, S_{k}\}$ be a $k$-partition of $V(G)$. The representation $r(v|\Pi)$ of a vertex $v$ with respect to $\Pi$ is the vector $(d(v, S_{1}), d(v, S_2), \dots, d(v, S_{k}))$, where $d(v, S_{i}) = \min\{d(v, s_{i})\mid s_{i}\in S_{i}\}$. The partition $\Pi$ is called a resolving partition of $G$ if $r(u|\Pi)\neq r(v|\Pi)$ for all distinct $u, v\in V(G)$. The partition dimension of $G$, denoted by $pd(G)$, is the cardinality of a minimum resolving partition of $G$. In this paper, we calculate the partition dimension of two $(4, 6)$-fullerene graphs. We also give conjectures on the partition dimension of two $(3, 6)$-fullerene graphs.

Sharp upper bounds on the Clar number of fullerene graphs

Sharp upper bounds on the Clar number of fullerene graphs

On the Eccentric Complexity of Graphs

Let G be a simple connected graph. The eccentric complexity of graph G is introduced as the number of different eccentricities of its vertices. A graph with eccentric complexity equal to one is called self-centered. In this paper, we study the eccentric complexity of graph under several graph operations such as complement of graph, line graph, Cartesian product, sum, disjunction and symmetric difference of graphs. Also, we present an infinite family of non-vertex-transitive self-centered graphs and we prove that all such graphs are 2-connected. Further, for any D and k where $$k \le \frac{D-1}{2}$$ , we construct an infinite family of non-vertex-transitive graphs with eccentric complexity k and diameter D. Extremal graphs with minimum or maximum total eccentricity among all graphs with given eccentric complexity are determined. We also consider a family of nanotubes and show that it is extremal with respect to the eccentric complexity among all fullerene graphs. At the end we also indicate some possible directions of further research.

Study of fullerenes via their symmetry groups

ABSTRACTIn this paper, we compute the automorphism group of some infinite classes of fullerene graphs and then we compute their symmetric Szeged index which is a new topological index based on the automorphism group.

Distance-restricted matching extendability of fullerene graphs

A fullerene graph is a 3-connected cubic plane graph whose all faces are bounded by 5- or 6-cycles. In this paper, we show that a matching M of a fullerene graph can be extended to a perfect matching if the following hold: (i) three faces around each vertex in \(\{x,y:xy\in M\}\) are bounded by 6-cycles and (ii) the edges in M lie at distance at least 13 pairwise.

On the maximum forcing and anti-forcing numbers of [formula omitted]-fullerenes

Let G be a (4,6)-fullerene graph. We show that the maximum forcing number of G is equal to its Clar number, and the maximum anti-forcing number of G is equal to its Fries number, which extend the known results for hexagonal systems with a perfect matching (Xu et al., 2013; Lei et al., 2016). Moreover, we obtain two formulas dependent only on the order of G to count the Clar number and Fries number of G respectively. Hence we can compute the maximum forcing number of a (4,6)-fullerene graph in linear time. This answers an open problem proposed by Afshani et al. (2004) in the case of (4,6)-fullerene graphs.

Leapfrog fullerenes and Wiener index

Fullerene graphs are cubic, 3-connected planar graphs with only pentagonal and hexagonal faces. A fullerene is called a leapfrog fullerene, Le(F), if it can be constructed by a leapfrog transformation from other fullerene graph F. Here we determine the relation between the Wiener index of Le(F) and the Wiener index of the original graph F. We obtain lower and upper bounds of the Wiener index of Lei(F) in terms of the Wiener index of the original graph. As a consequence, starting with any fullerene F, and iterating the leapfrog transformation we obtain fullerenes, Lei(F), with Wiener index of order O(n2.64) and Ω(n2.36), where n is the number of vertices of Lei(F). These results disprove Hua et al. (2014) conjecture that the Wiener index of fullerene graphs on n vertices is of order Θ(n3).

Classification of self-assembling protein nanoparticle architectures for applications in vaccine design.

We introduce here a mathematical procedure for the structural classification of a specific class of self-assembling protein nanoparticles (SAPNs) that are used as a platform for repetitive antigen display systems. These SAPNs have distinctive geometries as a consequence of the fact that their peptide building blocks are formed from two linked coiled coils that are designed to assemble into trimeric and pentameric clusters. This allows a mathematical description of particle architectures in terms of bipartite (3,5)-regular graphs. Exploiting the relation with fullerene graphs, we provide a complete atlas of SAPN morphologies. The classification enables a detailed understanding of the spectrum of possible particle geometries that can arise in the self-assembly process. Moreover, it provides a toolkit for a systematic exploitation of SAPNs in bioengineering in the context of vaccine design, predicting the density of B-cell epitopes on the SAPN surface, which is critical for a strong humoral immune response.

Relationship between Coefficients of Characteristic Polynomial and Matching Polynomial of Regular Graphs and its Applications

ABSTRACT. Suppose G is a graph, A(G) its adjacency matrix and f(G, x)=x^n+a_(n-1)x^(n-1)+... is the characteristic polynomial of G. The matching polynomial of G is defined as M(G, x) = x^n-m(G,1)x^(n-2) + ... where m(G,k) is the number of k-matchings in G. In this paper, we determine the relationship between 2k-th coefficient of characteristic polynomial, a_(2k), and k-th coefficient of matching polynomial, (-1)^km(G, k), in a regular graph. In the rest of this paper, we apply these relations for finding 5,6-matchings of fullerene graphs.

Mathematical aspects of fullerenes

Fullerene graphs are cubic, 3-connected, planar graphs with exactly 12 pentagonal faces, while all other faces are hexagons. Fullerene graphs are mathematical models of fullerene molecules, i.e., molecules comprised only by carbon atoms different than graphites and diamonds. We give a survey on fullerene graphs from our perspective, which could be also considered as an introduction to this topic. Different types of fullerene graphs are considered, their symmetries, and construction methods. We give an overview of some graph invariants that can possibly correlate with the fullerene molecule stability, such as: the bipartite edge frustration, the independence number, the saturation number, the number of perfect matchings, etc.

On the automorphism group of polyhedral graphs

A (4,6)-fullerene is a three connected cubic planar graph whose faces are squares and hexagons. In this paper, for a given (4,6)-fullerene graph F, we compute the order of automorphism group F. We also study some spectral properties of fullerene graphs via their automorphism group.

Comparing Fullerenes by Spectral Moments.

Suppose G is a graph, A(G) its adjacency matrix, and μ1(G)≤(G)μ2(G)≤ ... ≤ μ(n)(G)are eigenvalues of A(G). The numbers S(k)(G) = Σ(i) n = 1 μ(i)k (G), 0 ≤ k ≤ n -1 are said to be the k-th spectral moment of G and the sequence S(G) = (S0(G), S1 (G),..., S(n-1)(G)is called the spectral moments sequence of G. Suppose G1 and G2 are graphs. If there exists an integer k, 1 ≤ k ≤ n - 1, such that for each i, 0 ≤ i ≤ k - 1, S(i) (G1) = S(i)(G2) and S(k)(G1) < S(k)(G2) then we write G1 -<(s) G2. The aim of this paper is order some classes of fullerene graphs with respect to the S-order.

Matchings in m-generalized fullerene graphs

A connected planar graph is called m-generalized fullerene if two of its faces are m -gons and all other faces are pentagons and hexagons. In this paper we first determine some structural properties of m -generalized fullerenes and then use them to obtain new results on the enumerative aspects of perfect matchings in such graphs. We provide both upper and lower bounds on the number of perfect matchings in m -generalized fullerene graphs and state exact results in some special cases.

Bounds for the Sum-Balaban index and (revised) Szeged index of regular graphs

Mathematical properties of many topological indices are investigated. Knor et al. gave an upper bound for the Balaban index of r-regular graphs on n vertices and a better upper bound for fullerene graphs. They also suggested exploring similar bounds for other topological indices. In this paper, we consider the Sum-Balaban index and the (revised) Szeged index, and give upper and lower bounds for these three indices of r-regular graphs, and also the cubic graphs and fullerene graphs, respectively.

2-resonant fullerenes

A fullerene graph F is a planar cubic graph with exactly 12 pentagonal faces and other hexagonal faces. A set H of disjoint hexagons of F is called a resonant pattern (or sextet pattern) if F has a perfect matching M such that every hexagon in H is M-alternating. F is said to be k-resonant if any i (0≤i≤k) disjoint hexagons of F form a resonant pattern. It was known that each fullerene graph is 1-resonant and there are only nine fullerene graphs that are 3-resonant. In this paper, we show that the fullerene graphs which do not contain the subgraph L or R as illustrated in Fig. 1 are 2-resonant except for the specific eleven graphs. This result implies that each IPR fullerene is 2-resonant.

On the Topological Characterization of Fullerene-Like Materials Using Zagreb Indices-Based Topological Descriptors

To classify quantitatively the topological structure of all carbon fullerene molecules (fullerene graphs) we suggest a novel two-step method based on the following concept: As a first step, the duals of the fullerene graphs are generated, and as a second step, by using two molecular descriptors called Zagreb indices, we construct appropriately defined topological indices for structural characterization of fullerene isomers. Performing comparative tests on the set of C40 fullerene isomers, it will be demonstrated that the method suggested can be efficiently applicable to the stability prediction of fullerene-like materials.