厦门大学学报（自然科学版）2025，Vol.64Issue(6)：1011-1018,后插8-后插9,10.DOI:10.6043/j.issn.0438-0479.202411021
具有极值广义Clar覆盖多项式的六角链的刻画
Characterization of a class of hexagonal chain with extremal generalized Clar covering polynomials
摘要
Abstract
[Objective]The generalized Clar covering polynomial of hexagonal system contains many important issues in the Clar aromatic theory and the valence bond theory.These issues can be exemplified by the topological indicators,such as Kekulé number.Previous studies have shown that the generalized Clar covering polynomial can be used to accurately calculate the number of chemical activities including resonance energy.Resonant energy is often used to predict the aromatic stability of thick cyclic aromatic hydrocarbon conjugate systems.We give the characterization of hexagonal chains with the maximum and minimum generalized Clar covering polynomials,and present amendatory algorithm about computing the generalized Clar covering polynomial of arbitrary hexagonal chains.[Methods]First,the quasi-order of polynomial is used to explain the comparison of two Clar covering polynomials.Then we fix the number of hexagons of the hexagonal chain,and use mathematical induction to summarize what conditions the maximum linear chain of the hexagonal chain satisfies when its generalized Clar covering polynomial reaches its maximum.Finally,based on this finding,we split the maximal linear chain from different positions,observe and demonstrate the change of its generalized Clar covering polynomial.[Results]Induction results show that,when numbers of hexagons contained in each maximal linear chain are unified,the generalized Clar covering polynomial is greater than or equal to the generalized Clar covering polynomial of any hexagonal chain.And,the new hexagonal chain obtained after splitting the maximal linear chain from different positions,its generalized Clar covering polynomial is greater than or equal to the generalized Clar covering polynomial of the hexagonal chain that is not split.For any hexagonal chain with a fixed number of hexagons,to get a hexagonal chain with larger generalized Clar covering polynomial,just fission each of its maximal linear hexagonal chains.After a series of fissions,a hexagonal chain containing only two or three hexagons is obtained.Finally,we discuss the number of maximal linear hexagonal chains containing three hexagons and obtain the final result.Meanwhile,we propose an amendatory algorithm for calculating the generalized Clar covering polynomial for any hexagonal chains according to the existing recursive formulas.According to the algorithm,we can obtain the generalized Clar covering polynomials for the hexagonal chain of hexagonal numbers m=2;3;4;5;6 and find that the generalized Clar covering polynomial reaches its minimum on a linear chain and reaches its maximum on the ZigZag hexagonal chain.[Conclusions]Through algorithm examples and proofs,we have found interesting behaviors of any hexagonal chain with a fixed number of hexagons.When each maximal linear chain contains only two hexagons(i.e.ZigZag hexagonal chain),its generalized Clar covering polynomial reaches its maximum.On the contrary,the generalized Clar covering polynomial of a hexagonal chain containing only one maximal linear hexagonal chain reaches its minimum.关键词
六角链/Clar覆盖多项式/广义Clar覆盖多项式Key words
hexagonal chain/Clar covering polynomial/generalized Clar covering polynomial分类
数理科学引用本文复制引用
严丹,边红,赵菲菲..具有极值广义Clar覆盖多项式的六角链的刻画[J].厦门大学学报（自然科学版）,2025,64(6):1011-1018,后插8-后插9,10.基金项目
国家自然科学基金(12361072) （12361072）
2023年新疆维吾尔自治区自然科学基金面上项目(2023D01A36)和青年项目(2023D01B48) （2023D01A36）
国家自然科学基金数学天元基金(12426105) （12426105）
2022年新疆师范大学创新教学团队项目 （）
