具有完美匹配的图依能量的排序
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摘要
图的能量定义为图连接矩阵的所有特征根的绝对值之和．在化学图论中，研究具有极值能量的图具有十分重要的理论意义和应用价值．图的能量越大(小)，相应化合物的热力学稳定性越强(弱)．
共轭分子图根据其是否具有Kekuléan结构，可分为Kekuléan分子图和非Kekuléan分子图．在图论中，Kekuléan结构称为完美匹配．具有完美匹配的图具有许多化学的性质．例如，是否具有完美匹配对于芳香族系统的稳定性是及其重要的．
基于具有完美匹配的图在化学上的重要性，本文研究此类图的极值能量图问题．记此类图的顶点个数为2n．主要工作和结果有以下五部分．
(1)．考虑了最大度不超过3，并且具有完美匹配的树依能量从小到大的排序问题．我们采用了比文献(Li H．J．Math．Chem．25(1999)145-169)更加简捷的方法，当n+1≥14时，得到了2n-2r-5个具有较小能量的树并加以排序，其中r由n+1≡r(mod 4)确定．这个结果比文献(Li 1999)中得到的具有较小能量树的个数多了n-r-6个．当6≤n+1≤13时，我们也分别得到了许多具有较小能量的树并加以排序．
(2)．考虑了直径为d，并且具有完美匹配的树的极值能量图问题．当4≤d≤10时，分别得到了极小能量图．对于d=5，当n=2h和n=2h+1时，还分别得到了(1+(?))／2和(?)+1个具有较大能量的树并进行了排序，其中h是不小于2的正整数．
(3)．考虑了具有完美匹配的树依能量从小到大的排序问题．我们运用了比文献(ZhangF．J．&Li H．Discrete Appl．Math．92(1999)71-84)更加简单的证明方法，得到了此类图的极小，次二小和次三小能量图．
(4)．考虑了最大度不超过3，并且具有完美匹配的单圈图依Hosoya指标的排序及其极小能量图问题．首先，在四种特殊情况下，得到了这四类图依Hosoya指标从小到大的排序．接着，确定了所考虑图类中多个具有较小Hosoya指标的单圈图并加以排序．进一步地，得到了此类图的极小能量图．
(5)．考虑了具有完美匹配的单圈图，得到了此类图的极小能量图．
The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. The investigation on the graphs with extremal energies is of theoretical interest and practical importance in the subject of chemical graph theory. The larger the value of enery, the greater the thermodynamic stability of the corresponding compund.
Conjugated molecules in chemistry may be classified into two groups: Kekulean and non-Kekulean molecules, depending on whether or not they possess Kekulean structures, i.e., the perfect matchings in graph theory. The graphs with perfect matchings possess many chemical properties. For example, the existence of perfect matchings is closely connected to the stability of aromatic systems.
In view of the significance of the graphs with perfect mathings in chemical graph theory, this kind of the graphs with extremal energies is investigated in this thesis. The vertex number of the graphs considered is denoted by In. The main results can be divided into five parts as follows.
Firstly, the ordering of the minimal energies of the trees with a perfect matching having degrees no greater than three is considered. By means of a simpler method than that of Li (J. Math. Chem. 25 (1999) 145-169), we obtain the first 2n - 2r - 5 trees in the increasing order of their energies within the class under consideration for n + 1≥14, where r is determined by n + 1≡r (mod 4). The number of the trees obtained here exceeds the reported result (Li 1999) by n - r - 6. We also get a lot of preceding trees in the increasing order of their energies within the class for 6≤n + 1≤13.
Secondly, the extremal energies of the trees with a perfect matching having a given diameter d are obtained. The graphs with minimal energies are given for 4≤d≤10. As d = 5, we obtain the last (1 +(?))/2 and y(?)+1 trees in the increasing order of their energies within the class under consideration for n = 2h and n = 2h + 1, respectively, where h≥2.
Thirdly, the ordering of the minimal energies of the trees with a perfect matching is studied. We employ a simpler method than that of Zhang & Li (Discrete Appl. Math. 92 (1999) 71-84) to find the trees having the minimal, the second-minimal, and the third-minimal energies.
Next, the ordering of unicyclic graphs with perfect matchings having degrees no greater than three by their Hosoya indexes is considered. Four special cases in the increasing order of their Hosoya indexes are studied. The preceding graphs in the increasing order of their Hosoya indexes are determined. Furthermore, the corresponding graphs with minimal energies are provided.
Finally, unicyclic graphs with perfect matchings having minimal energy are considered . The corresponding graph with minimal energy is mathematically verified.
共轭分子图根据其是否具有Kekuléan结构，可分为Kekuléan分子图和非Kekuléan分子图．在图论中，Kekuléan结构称为完美匹配．具有完美匹配的图具有许多化学的性质．例如，是否具有完美匹配对于芳香族系统的稳定性是及其重要的．
基于具有完美匹配的图在化学上的重要性，本文研究此类图的极值能量图问题．记此类图的顶点个数为2n．主要工作和结果有以下五部分．
(1)．考虑了最大度不超过3，并且具有完美匹配的树依能量从小到大的排序问题．我们采用了比文献(Li H．J．Math．Chem．25(1999)145-169)更加简捷的方法，当n+1≥14时，得到了2n-2r-5个具有较小能量的树并加以排序，其中r由n+1≡r(mod 4)确定．这个结果比文献(Li 1999)中得到的具有较小能量树的个数多了n-r-6个．当6≤n+1≤13时，我们也分别得到了许多具有较小能量的树并加以排序．
(2)．考虑了直径为d，并且具有完美匹配的树的极值能量图问题．当4≤d≤10时，分别得到了极小能量图．对于d=5，当n=2h和n=2h+1时，还分别得到了(1+(?))／2和(?)+1个具有较大能量的树并进行了排序，其中h是不小于2的正整数．
(3)．考虑了具有完美匹配的树依能量从小到大的排序问题．我们运用了比文献(ZhangF．J．&Li H．Discrete Appl．Math．92(1999)71-84)更加简单的证明方法，得到了此类图的极小，次二小和次三小能量图．
(4)．考虑了最大度不超过3，并且具有完美匹配的单圈图依Hosoya指标的排序及其极小能量图问题．首先，在四种特殊情况下，得到了这四类图依Hosoya指标从小到大的排序．接着，确定了所考虑图类中多个具有较小Hosoya指标的单圈图并加以排序．进一步地，得到了此类图的极小能量图．
(5)．考虑了具有完美匹配的单圈图，得到了此类图的极小能量图．
The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. The investigation on the graphs with extremal energies is of theoretical interest and practical importance in the subject of chemical graph theory. The larger the value of enery, the greater the thermodynamic stability of the corresponding compund.
Conjugated molecules in chemistry may be classified into two groups: Kekulean and non-Kekulean molecules, depending on whether or not they possess Kekulean structures, i.e., the perfect matchings in graph theory. The graphs with perfect matchings possess many chemical properties. For example, the existence of perfect matchings is closely connected to the stability of aromatic systems.
In view of the significance of the graphs with perfect mathings in chemical graph theory, this kind of the graphs with extremal energies is investigated in this thesis. The vertex number of the graphs considered is denoted by In. The main results can be divided into five parts as follows.
Firstly, the ordering of the minimal energies of the trees with a perfect matching having degrees no greater than three is considered. By means of a simpler method than that of Li (J. Math. Chem. 25 (1999) 145-169), we obtain the first 2n - 2r - 5 trees in the increasing order of their energies within the class under consideration for n + 1≥14, where r is determined by n + 1≡r (mod 4). The number of the trees obtained here exceeds the reported result (Li 1999) by n - r - 6. We also get a lot of preceding trees in the increasing order of their energies within the class for 6≤n + 1≤13.
Secondly, the extremal energies of the trees with a perfect matching having a given diameter d are obtained. The graphs with minimal energies are given for 4≤d≤10. As d = 5, we obtain the last (1 +(?))/2 and y(?)+1 trees in the increasing order of their energies within the class under consideration for n = 2h and n = 2h + 1, respectively, where h≥2.
Thirdly, the ordering of the minimal energies of the trees with a perfect matching is studied. We employ a simpler method than that of Zhang & Li (Discrete Appl. Math. 92 (1999) 71-84) to find the trees having the minimal, the second-minimal, and the third-minimal energies.
Next, the ordering of unicyclic graphs with perfect matchings having degrees no greater than three by their Hosoya indexes is considered. Four special cases in the increasing order of their Hosoya indexes are studied. The preceding graphs in the increasing order of their Hosoya indexes are determined. Furthermore, the corresponding graphs with minimal energies are provided.
Finally, unicyclic graphs with perfect matchings having minimal energy are considered . The corresponding graph with minimal energy is mathematically verified.
引文
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[7] Narumi H. & Hosoya H. Topological index and thermodynamic properties. III. Classification of various topological aspects of properties of acyclic saturated hydrocarbons. Bull. Chem. Soc. Japan 58 (1985) 1778-1786.
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[12] Gutman I. Note on the Coulson integral formula. J. Math. Chem. 39 (2005) 259-266.
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[2] Gutman I. & Polansky O.E. Mathematical Concepts in Organic Chemistry. Berlin:Springer-Verlag (1986).
[3] Gutman I. The energy of a graph: old and new results, in: A. Betten, A. Kohnert,R. Laue, A. Wassermann (Eds.) Algebra Combinatorics and Applications, pp. 196-211.Berlin: Springer-Verlag (2001).
[4] Gutman I. Topology and stability of conjugated hydrocarbons. The dependence of total7r-electron energy on molecular topology. J. Serb. Chem. Soc. 70 (2005) 441-456.
[5]唐熬庆，江元声等．分子轨道图形理论．北京：科学出版社(1980)．
[6] Bonchev D. & Rouvray D.H. Chemical Graph Theory: Introduction and Fundamentals.Gordon and Breach Science Pulishers (1990).
[7] Narumi H. & Hosoya H. Topological index and thermodynamic properties. III. Classification of various topological aspects of properties of acyclic saturated hydrocarbons. Bull. Chem. Soc. Japan 58 (1985) 1778-1786.
[8] Gutman I. & Zhang F. J. On the ordering of graphs with respect to their matching numbersDiscrete Appl. Math. 15 (1986) 25-33.
[9] Hosoya H. Introduction to Graph Theory. In: D. Bonchev and O. Mekenyan ed. Graph Theoretical Approaches to Chemical Reactivity, p. 30, Kluwer Academic Publishers (1994).
[10] Trinajstic N. Chemical Graph Theory, Vol. 2. CRC Press (1983).
[11] Gutman I. Total 7r-electron energy of benzenoid hydrocarbon, in: Topics In Curr. Chem. Vol. 162: Advance in the Theory of Benzenoid Hydrocarbons II. pp. 29-63, Berlin: Springer (1992).
[12] Gutman I. Note on the Coulson integral formula. J. Math. Chem. 39 (2005) 259-266.
[13] Cvetkovic D.M., Doob M. & Sachs H. Spectra of Graphs Theory and Application. New York: Academic Press (1980).
[14] Hall G.G. The bond orders of alternant hydrocarbon molecules. Proc. Roc. Soc. London A 229 (1955) 251-259.
[15] McClelland B.J. Properties of latent roots of a matrix-estimation of electron energies. J. Chem. Phys. 54 (1971) 640-643.
[16] England W. & Ruedenberg K. Why is delocalization energy negative and why is it proportional to number of electron. J. Amer. Chem. Soc. 95 (1973) 8769-8775.
[17] Gutman I., Nedeljkovic L.J. & Teodorovic A.V. Bull. Soc. Chim. Beograd 48 (1983) 495.
[18] Gutman I., Soldatovic T. k Vidovic D. The energy of a graph and its size dependence. AMonte Carlo approach. Chem. Phys. Lett. 297 (1998) 428-432.
[19] Gutman I. Bounds for total 7r-electron energy. Chem. Phys. Lett. 24 (1974) 283-285.
[20] Gutman I. Bounds for total 7r-electron energy of polymethines. Chem. Phys. Lett. 50(1977) 488-490.
[21] Gutman I. Bounds for total 7r-electron energy. Croat. Chem. Ada 51 (1978) 299-306.
[22] Gutman I. Acyclic Conjugated molecules, trees and their energies. J. Math. Chem. 1(1987) 123-143.
[23] Turker L. An upper bound for total 7r-electron energy of alternant hydrocarbons. Commun.Math. Chem. 6 (1984) 83-94.
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