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The Structural Properties of (2,6)-Fullerenes
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: Received: 20 September 2023 / Approved: 20 September 2023 / Online: 25 September 2023 (08:05:56 CEST)
A peer-reviewed article of this Preprint also exists.
Yang, R.; Yuan, M. The Structural Properties of (2, 6)-Fullerenes. Symmetry 2023, 15, 2078. Yang, R.; Yuan, M. The Structural Properties of (2, 6)-Fullerenes. Symmetry 2023, 15, 2078.
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 Do\v{s}li\'{c}'s (k,6)-fullerene, a 2-connected 3-regular plane graph with only k-length faces and hexagons. Do\v{s}li\'{c} 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 are studied. In this paper, we study the properties of (2,6)-fullerene. We obtain that the edge-connectivity of (2,6)-fullerenes is 2. Every (2,6)-fullerene is 1-extendable, but not 2-extendable. F is said to be k- resonant (k>0) 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-Kekule 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-Kekule number of F, denoted by ak(F), is the cardinality of a smallest anti-Kekule set of F. We have that every (2,6)-fullerene F with |V(F)|>6 has anti-Kekule 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, m.
Keywords
(2,6)-fullerene; edge-connectivity; anti-Kekule number; resonance; perfect matching
Subject
Computer Science and Mathematics, Discrete Mathematics and Combinatorics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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