Total Irregularity Strengths of an Arbitrary Disjoint Union of (3,6)- Fullerenes

Aims and Objective: A fullerene graph is a mathematical model of a fullerene molecule. A fullerene molecule or simply a fullerene is a polyhedral molecule made entirely of carbon atoms other than graphite and diamond. Chemical graph theory is a combination of chemistry and graph theory where graph theoretical concepts used to study physical properties of mathematically modeled chemical compounds. Graph labeling is a vital area of graph theory which has application not only within mathematics but also in computer science, coding theory, medicine, communication networking, chemistry and in many other fields. For example, in chemistry vertex labeling is being used in the constitution of valence isomers and transition labeling to study chemical reaction networks. Method and Results: In terms of graphs vertices represent atoms while edges stand for bonds between atoms. By tvs (tes) we mean the least positive integer for which a graph has a vertex (edge) irregular total labeling such that no two vertices (edges) have same weights. A (3,6)-fullerene graph is a non-classical fullerene whose faces are triangles and hexagons. Here, we study the total vertex (edge) irregularity strength of an arbitrary disjoint union of (3,6)-fullerene graphs and providing their exact values. Conclusion: The lower bound for tvs (tes) depending on the number of vertices, minimum and maximum degree of a graph exists in literature while to get different weights one can use sufficiently large numbers, but it is of no interest. Here, by proving that the lower bound is the upper bound we close the case for (3,6)-fullerene graphs.

All Pairs of Pentagons in Leapfrog Fullerenes Are Nice

A subgraph H of a graph G with perfect matching is nice if G−V(H) has perfect matching. It is well-known that all fullerene graphs have perfect matchings and that all fullerene graphs contain some small connected graphs as nice subgraphs. In this contribution, we consider fullerene graphs arising from smaller fullerenes via the leapfrog transformation, and show that in such graphs, each pair of (necessarily disjoint) pentagons is nice. That answers in affirmative a question posed in a recent paper on nice pairs of odd cycles in fullerene graphs.

Properties of Entropy-Based Topological Measures of Fullerenes

A fullerene is a cubic three-connected graph whose faces are entirely composed of pentagons and hexagons. Entropy applied to graphs is one of the significant approaches to measuring the complexity of relational structures. Recently, the research on complex networks has received great attention, because many complex systems can be modelled as networks consisting of components as well as relations among these components. Information—theoretic measures have been used to analyze chemical structures possessing bond types and hetero-atoms. In the present article, we reviewed various entropy-based measures on fullerene graphs. In particular, we surveyed results on the topological information content of a graph, namely the orbit-entropy Ia(G), the symmetry index, a degree-based entropy measure Iλ(G), the eccentric-entropy Ifσ(G) and the Hosoya entropy H(G).

THE MOSTAR INDEX OF FULLERENES IN TERMS OF AUTOMORPHISM GROUP

Let $G$ be a connected graph. For an edge $e=uv\in E(G)$, suppose $n(u)$ and $n(v)$ are respectively, the number of vertices of $G$ lying closer to vertex $u$ than to vertex $v$ and the number of vertices of $G$ lying closer to vertex $v$ than to vertex $u$. The Mostar index is a topological index which is defined as $Mo(G)=\sum_{e\in E(G)}f(e)$, where $f(e) = |n(u)-n(v)|$. In this paper, we will compute the Mostar index of a family of fullerene graphs in terms of the automorphism group.