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- Title
The Structural Properties of (2, 6)-Fullerenes.
- Authors
Yang, Rui; Yuan, Mingzhu
- Abstract
A (2 , 6) -fullerene F is a 2-connected cubic planar graph whose faces are only 2 -length and 6 -length. Furthermore, it consists of exactly three 2 -length faces by Euler's formula. The (2 , 6) -fullerene comes from Došlić's (k , 6) -fullerene, a 2-connected 3-regular plane graph with only k -length faces and hexagons. Došlić showed that the (k , 6) -fullerenes only exist for k = 2 , 3, 4, or 5, and some of the structural properties of (k , 6) -fullerene for k = 3 , 4, or 5 were studied. The structural properties, such as connectivity, extendability, resonance, and anti−Kekulé number, are very useful for studying the number of perfect matchings in a graph, and thus for the study of the stability of the molecular graphs. In this paper, we study the properties of (2 , 6) -fullerene. We discover that the edge-connectivity of (2 , 6) -fullerenes is 2. Every (2 , 6) -fullerene is 1-extendable, but not 2-extendable (F is called n - e x t e n d a b l e ( | V (F) | ≥ 2 n + 2 ) if any matching of n edges is contained in a perfect matching of F). F is said to be k-resonant ( k ≥ 1 ) if the deleting of any i ( 0 ≤ i ≤ k ) disjoint even faces of F results in a graph with at least one perfect matching. We have that every (2 , 6) -fullerene is 1-resonant. An edge set, S, of F is called an anti−Kekulé set if F − S is connected and has no perfect matchings, where F − S denotes the subgraph obtained by deleting all edges in S from F. The anti−Kekulé number of F, denoted by a k (F) , is the cardinality of a smallest anti−Kekulé set of F. We have that every (2 , 6) -fullerene F with | V (F) | > 6 has anti−Kekulé number 4. Further we mainly prove that there exists a (2 , 6) -fullerene F having f F hexagonal faces, where f F is related to the two parameters n and m.
- Subjects
MOLECULAR graphs; PLANAR graphs; MATCHING theory; NUMBER theory; RESONANCE; RAMSEY numbers; HEXAGONS
- Publication
Symmetry (20738994), 2023, Vol 15, Issue 11, p2078
- ISSN
2073-8994
- Publication type
Article
- DOI
10.3390/sym15112078