随机环辛烷链的三类Kirchhoff指数

刘合超; 吴让威; 尤利华

doi:10.6054/j.jscnun.2021031

随机环辛烷链的三类Kirchhoff指数

华南师范大学数学科学学院，广州 510631

基金项目:

国家自然科学基金项目 11971180

国家自然科学基金项目 11571123

广东省自然科学基金项目 2019A1515012052

详细信息

通讯作者:
尤利华, Email: ylhua@scnu.edu.cn

中图分类号: O157.5
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出版历程
- 收稿日期: 2020-09-28
- 网络出版日期: 2021-04-28
- 刊出日期: 2021-04-24

Three Types of Kirchhoff Indices in the Random Cyclooctane Chains

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

摘要

摘要: 利用递归的方法, 首先确定了随机环辛烷链的3类Kirchhoff指数(Kirchhoff指数、度积Kirchhoff指数、度和Kirchhoff指数)的期望表达式, 再由该式得到了具有n个八边形的环辛烷链的3类Kirchhoff指数的极大值与极小值, 并刻画了相应的极图.
- 随机环辛烷链 /
- 期望 /
- Kirchhoff指数 /
- 度积Kirchhoff指数 /
- 度和Kirchhoff指数
Abstract: Using the recursive method, the formula for the expected values of three types of Kirchhoff indices (Kirchhoff index, multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index) in the random cyclooctane chains is first determined. Then the maximum and minimum values of three types of Kirchhoff indices among cyclooctane chains with n octagons (based on this formula) is obtained. All the corresponding extremal graphs are characterized.
- random cyclooctane chains /
- expected value /
- Kirchhoff index /
- multiplicative degree-Kirchhoff index /
- additive degree-Kirchhoff index

HTML全文

拓扑指数是一类刻画分子结构的数值不变量. 在理论化学中, 拓扑指数被用于预测化合物的物理化学性质、药理活性和生理活性等. 常用的与距离有关的拓扑指数主要有Wiener指数^[1-2]、Harary指数^[3]和Kirchhoff指数^[4].

环辛烷及其衍生物是有机化学中重要的环烷烃, 可用于药物合成、有机合成等. 而研究一些特殊化学链的拓扑指数及其在化学中的应用也成为化学图论的热点问题^[5-13]. 如：WU等^[5]计算了聚苯链、聚苯蜘蛛链的hyper-Wiener指数, 得到上、下界; YANG和ZHANG^[6]给出了随机聚苯链的Wiener指数的期望值; WEI等^[12]给出了随机环辛烷链的Wiener指数的期望值; ZHANG等^[13]确定了随机聚苯链的Schultz指数、Gutman指数、度积Kirchhoff指数、度和Kirchhoff指数的期望值.

关于Kirchhoff指数的研究已经有比较多的成果. 如：YANG和JIANG^[14]给出了单圈图的Kirchhoff指数的极大、极小值; GUO等^[15]确定了全负荷单圈图的Kirchhoff指数的极大、极小值; FENG等^[16]刻画了具有极大、极小度Kirchhoff指数的全负荷单圈图; FEI和TU^[17]刻画了具有最大、第二大度Kirchhoff指数的双圈图. 关于更多介绍Kirchhoff指数的性质及其应用的文献可参考文献[18-21]. 在文献[12-13]的启发下, 本文首先确定了随机环辛烷链的3类Kirchhoff指数(Kirchhoff指数、度积Kirchhoff指数、度和Kirchhoff指数)的期望表达式, 再由该式得到了具有n个八边形的环辛烷链的3类Kirchhoff指数的极大值与极小值, 并刻画了相应的极图.

1. 预备知识

1947年, WIENER^[1]在研究石蜡的沸点与其分子的结构关系时, 提出了用拓扑指数来研究分子的结构性质. 由于当时化学图论发展的限制, H WIENER并未将这一指数与化学图论结合给出定义. 1971年, HOSOYA^[2]对这一指数给出了图论术语表达式, 即 $W\left( G \right) = \sum\limits_{\left\{ {u, v} \right\} \subseteq V\left( G \right)} {{d_G}\left( {u, v} \right)}$ , 其中d_G(u, v)表示顶点u、v之间最短路的长度. 为了纪念H WIENER对这一指数的贡献, 将其称为Wiener指数.

1993年, KLEIN和RANDI Ć^[22]基于电网理论引入了一个新的距离函数, 称之为电阻距离: 通过用单位电阻器代替图G的每条边来将图G视为电网N, 将电网N中顶点u与顶点v之间的有效电阻值称为图G中顶点u与顶点v之间的电阻距离r_G(u, v).

在Wiener指数的定义中用电阻距离替代距离, 则可得到Kirchhoff指数^[4]：

$\operatorname{Kf}(G)=\sum\limits_{\{u, v\}\subseteq V(G)} r_{G}(u, v).$

度积Kirchhoff指数Kf^*(G)^[23]与度和Kirchhoff指数Kf ⁺(G)^[24]是2类加权版的Kirchhoff指数, 分别定义为:

$\begin{aligned} \mathrm{Kf}^{*}(G) &=\sum\limits_{\{u, v\}\subseteq V(G)}\left(d_{G}(u) d_{G}(v)\right) r_{G}(u, v), \\ \mathrm{Kf}^{+}(G) &=\sum\limits_{\{u, v\} \subseteq V(G)}\left(d_{G}(u)+d_{G}(v)\right) r_{G}(u, v) . \end{aligned}$

图 1为一条具有n个正八边形的环辛烷链COC_n, 可以看成是将环辛烷链COC_n-1通过一条割边连接一个新的终端正八边形O_n所构成的.

图 1 n个正八边形的环辛烷链COC_n

Figure 1. A cyclooctane chain with n regular octagons

下载: 全尺寸图片幻灯片

设COC_n=O₁O₂…O_n为具有n (n≥2)个正八边形的环辛烷链, 其中O_t (t=2, 3, …, n)为图COC_n中通过割边u_t-1v_t与O_t-1相连的第t个正八边形. 在O_t中, 与点v_t距离为1、2、3、4的点分别记为o_t、m_t、w_t、l_t. 例如: 图 1中v_n=z₁; o_n=z₂, z₈; m_n=z₃, z₇; w_n=z₄, z₆; l_n=z₅.

在具有n个正八边形的环辛烷链COC_n中, 有一些特殊类的COC_n. 例如: 对任意的2≤t≤n-1, 若均有u_t=o_t, 则将该环辛烷链COC_n记为CO_n; 若u_t=m_t, 则将其记为CM_n; 如u_t=w_t, 则将其记为CW_n; 若u_t=l_t, 则将其记为CL_n.

当n≥3时, COC_n+1的终端八边形O_n+1可以连接到O_n的o_n、m_n、w_n、l_n点, 即4种连接方式, 不妨将得到的4种不同的图形(同构意义下)分别记为COC_n+1¹、COC_n+1²、COC_n+1³和COC_n+1⁴, 见图 2.

图 2 环辛烷链的4种连接方式

Figure 2. Four connection methods of cyclooctane chains

下载: 全尺寸图片幻灯片

一条具有n个正八边形的随机环辛烷链, 记为COC(n; p₁, p₂, p₃), 是按照以下的连接方式进行n-2步得到的链(每一步t (t=3, 4, …, n)都随机选择其中一种连接方式)：(1)COC_t-1→COC_t¹, 可能性为p₁; (2)COC_t-1→COC_t², 可能性为p₂; (3)COC_t-1→COC_t³, 可能性为p₃; (4)COC_t-1→COC_t⁴, 可能性为1-p₁-p₂-p₃, 其中p₁、p₂、p₃是与参数t无关的常数, 且0≤p₁, p₂, p₃≤1, 0≤p₁+p₂+p₃≤1.

特别地, COC(n; 1, 0, 0)、COC(n; 0, 1, 0)、COC(n; 0, 0, 1)、COC(n; 0, 0, 0)分别为链CO_n、CM_n、CW_n、CL_n.

2. 随机环辛烷链的Kirchhoff指数

用E[Kf(COC(n; p₁, p₂, p₃))]表示随机环辛烷链COC(n; p₁, p₂, p₃)的Kirchhoff指数的期望, 记r(u|G)= $\sum\limits_{v \in V\left( G \right)} {r\left( {u, v} \right)}$ . 为了计算随机环辛烷链的Kirchhoff指数期望的表达式, 首先给出COC(n; p₁, p₂, p₃)的Kirchhoff指数的递归公式.

引理1 设COC_n为具有n (n≥1)个正八边形的环辛烷链, 则

$\begin{array}{l} \mathrm{Kf}\left(\mathrm{COC}_{n}\right)=\mathrm{Kf}\left(\mathrm{COC}_{n-1}\right)+8 r\left(u_{n-1} \mid \mathrm{COC}_{n-1}\right)+ \\ \quad\quad 148(n-1)+42 . \end{array}$

(1)

证明首先介绍如何求图中两顶点之间的Kirchhoff指数, 以图 1中顶点u_n-1与顶点z₇之间的Kirchhoff指数为例. 由物理中电阻串联、并联的规律知, 2个电阻值分别为R₁和R₂的电阻串联的总电阻为R₁+R₂, 并联的总电阻为R₁R₂/(R₁+R₂). 由图 1可以看出顶点u_n-1到顶点z₇的连接情况为：路z₁z₂z₃z₄z₅z₆z₇与路z₁z₈z₇并联, 然后与路z₁u_n-1串联. 从而可得顶点u_n-1与顶点z₇之间的Kirchhoff指数为 $\frac{{2 \times 6}}{{2 + 6}} + 1 = \frac{5}{2}$ . 下面对COC_n中的顶点进行分类，以求解任意顶点对x、y的电阻距离r(x, y).

对于任意顶点v∈V(COC_n-1), 由图 1知r(z_i, v)=r(z_i, u_n-1)+r(u_n-1, v), 且

$r\left(z_{i}, v\right)=r\left(u_{n-1}, v\right)+\left\{\begin{array}{ll} 1 \quad(i=1) , \\ 1+\frac{7}{8}=\frac{15}{8} \quad (i=2,8), \\ 1+\frac{3}{2}=\frac{5}{2} \quad (i=3,7), \\ 1+\frac{15}{8}=\frac{23}{8} \quad(i=4,6), \\ 1+2=3 \quad(i=5) . \end{array}\right.$

(2)

对于z_k ∈V(O_n) (1≤k≤8), 有

$\sum\limits_{i=1}^{8} r\left(z_{k}, z_{i}\right)=\left(\frac{7}{8}+\frac{3}{2}+\frac{15}{8}\right) \times 2+2=\frac{21}{2}.$

由r(x|G)= $\sum\limits_{y \in V\left( G \right)} {r\left( {x, y} \right)}$ 及式(2), 有

$\begin{array}{c} r\left(z_{i} \mid \mathrm{COC}_{n}\right)=\sum\limits_{j=1}^{8} r\left(z_{i}, z_{j}\right)+\sum\limits_{v \in V\left(\operatorname{COC}_{n-1}\right)} r\left(z_{i}, v\right)= \\ \frac{21}{2}+8(n-1) r\left(z_{i}, u_{n-1}\right)+r\left(u_{n-1} \mid \mathrm{COC}_{n-1}\right), \end{array}$

从而

$r\left(z_{i} \mid \mathrm{COC}_{n}\right)=r\left(u_{n-1} \mid \mathrm{COC}_{n-1}\right)+\left\{\begin{array}{ll} 8 n+\frac{5}{2} & (i=1), \\ 15 n-\frac{9}{2} & (i=2,8), \\ 20 n-\frac{19}{2} & (i=3,7), \\ 23 n-\frac{25}{2} & (i=4,6), \\ 24 n-\frac{27}{2} & (i=5) . \end{array}\right.$

(3)

进而

$\begin{array}{c} \mathrm{Kf}\left(\mathrm{COC}_{n}\right)=\mathrm{Kf}\left(\mathrm{COC}_{n-1}\right)+\sum\limits_{i=1}^{8} r\left(z_{i} \mid \mathrm{COC}_{n}\right)- \\ \frac{1}{2} \sum\limits_{i=1}^{8} \sum\limits_{j=1}^{8} r\left(z_{i}, z_{j}\right)=\mathrm{Kf}\left(\mathrm{COC}_{n-1}\right)+ \\ 8 r\left(u_{n-1} \mid \mathrm{COC}_{n-1}\right)+148(n-1)+42. \end{array}$

故式(1)成立. 证毕.

记U_n=E[r(u_n|COC_n)], 下面给出U_n的表达式.

引理2 设COC(n; p₁, p₂, p₃)为具有n (n≥1)个正八边形的随机环辛烷链, 则

$U_{n}=\left(12-\frac{9}{2} p_{1}-2 p_{2}-\frac{1}{2} p_{3}\right) n^{2}+\left(\frac{9}{2} p_{1}+2 p_{2}+\frac{1}{2} p_{3}-\frac{3}{2}\right) n.$

(4)

证明分以下4种情况来计算U_n.

情形1：COC_n→COC_n+1¹. 此时u_n=z₂或z₈, r(u_n|COC_n)为r(z₂|COC_n)的可能性为p₁.

情形2：COC_n→COC_n+1². 此时u_n=z₃或z₇, r(u_n|COC_n)为r(z₃|COC_n)的可能性为p₂.

情形3：COC_n→COC_n+1³. 此时u_n=z₄或z₆, r(u_n|COC_n)为r(z₄|COC_n)的可能性为p₃.

情形4：COC_n→COC_n+1⁴. 此时u_n=z₅, r(u_n|COC_n)为r(z₅|COC_n)的可能性为1-p₁-p₂-p₃.

由情形1至情形4及式(3), 可知

$\begin{aligned} U_{n}=& r\left(z_{2} \mid \mathrm{COC}_{n}\right) p_{1}+r\left(z_{3} \mid \mathrm{COC}_{n}\right) p_{2}+r\left(z_{4} \mid \mathrm{COC}_{n}\right) p_{3}+ \\ & r\left(z_{5} \mid \mathrm{COC}_{n}\right)\left(1-p_{1}-p_{2}-p_{3}\right)= \\ & p_{1}\left[r\left(u_{n-1} \mid \operatorname{COC}\left(n-1 ; p_{1}, p_{2}, p_{3}\right)\right)+15 n-\frac{9}{2}\right]+ \\ & p_{2}\left[r\left(u_{n-1} \mid \operatorname{COC}\left(n-1 ; p_{1}, p_{2}, p_{3}\right)\right)+20 n-\frac{19}{2}\right]+ \\ & p_{3}\left[r\left(u_{n-1} \mid \operatorname{COC}\left(n-1 ; p_{1}, p_{2}, p_{3}\right)\right)+23 n-\frac{25}{2}\right]+ \\ & \left(1-p_{1}-p_{2}-p_{3}\right)\left[r\left(u_{n-1} \mid \operatorname{COC}\left(n-1 ; p_{1}, p_{2}, p_{3}\right)\right)+\right. \\ & \left.24 n-\frac{27}{2}\right]. \end{aligned}$

而E[U_n]=U_n, 则

$\begin{aligned} U_{n}&=p_{1}\left[U_{n-1}+15 n-\frac{9}{2}\right]+p_{2}\left[U_{n-1}+20 n-\frac{19}{2}\right]+ \\ &p_{3}\left[U_{n-1}+23 n-\frac{25}{2}\right]+\left(1-p_{1}-p_{2}-p_{3}\right)\left[U_{n-1}+24 n-\frac{27}{2}\right]= \\ &U_{n-1}+\left(24-9 p_{1}-4 p_{2}-p_{3}\right) n+\left(9 p_{1}+4 p_{2}+p_{3}-\frac{27}{2}\right). \end{aligned}$

(5)

将初始值U₁=E[r(u₁|COC(1;p₁, p₂, p₃))]=( $\frac{7}{8} + \frac{3}{2} + \frac{{15}}{8}$ )×2+2= $\frac{{21}}{2}$ 代入式(5), 递归可得式(4). 证毕.

由引理1和引理2, 易得随机环辛烷链的Kirchhoff指数的期望值.

定理1 设COC(n; p₁, p₂, p₃)为具有n (n≥1)个正八边形的随机环辛烷链, 则

$\begin{array}{l} E\left[\operatorname{Kf}\left(\operatorname{COC}\left(n ; p_{1}, p_{2}, p_{3}\right)\right)\right]= \\ \quad \frac{1}{3}\left(96-36 p_{1}-16 p_{2}-4 p_{3}\right) n^{3}+\left(20+36 p_{1}+16 p_{2}+4 p_{3}\right) n^{2}- \\ \quad \frac{1}{3}\left(30+72 p_{1}+32 p_{2}+8 p_{3}\right) n. \end{array}$

(6)

证明对式(1)两边取期望, 由式(4), 有

$\begin{array}{l} E\left[\operatorname{Kf}\left(\operatorname{COC}\left(n ; p_{1}, p_{2}, p_{3}\right)\right)\right]= \\ \quad E\left[\operatorname{Kf}\left(\operatorname{COC}\left(n-1 ; p_{1}, p_{2}, p_{3}\right)\right)\right]+ \\ \quad 8 E\left[r\left(u_{n-1} \mid \operatorname{COC}\left(n-1 ; p_{1}, p_{2}, p_{3}\right)\right)\right]+ \\ \quad 148(n-1)+42=E\left[\operatorname{Kf}\left(\operatorname{COC}\left(n-1 ; p_{1}, p_{2}, p_{3}\right)\right)\right]+ \\ \quad 8 U_{n-1}+148 n-106. \end{array}$

从而

$\begin{array}{l} E\left[\operatorname{Kf}\left(\operatorname{COC}\left(n+1 ; p_{1}, p_{2}, p_{3}\right)\right)\right]= \\ \quad E\left[\operatorname{Kf}\left(\operatorname{COC}\left(n ; p_{1}, p_{2}, p_{3}\right)\right)\right]+\left(96-36 p_{1}-16 p_{2}-4 p_{3}\right) n^{2}+ \\ \quad \left(136+36 p_{1}+16 p_{2}+4 p_{3}\right) n+42 . \end{array}$

(7)

将初始值E[Kf(COC(1;p₁, p₂, p₃))]=42代入式(7), 递归可得式(6)成立. 证毕.

由于COC(n; 1, 0, 0)、COC(n; 0, 1, 0)、COC(n; 0, 0, 1)、COC(n; 0, 0, 0)分别为链CO_n、CM_n、CW_n、CL_n. 将(p₁, p₂, p₃)=(1, 0, 0)、(p₁, p₂, p₃)=(0, 1, 0)、(p₁, p₂, p₃)=(0, 0, 1)、(p₁, p₂, p₃)=(0, 0, 0)分别代入定理1的期望表达式, 易得下面的推论.

推论1 4种特殊环辛烷链CO_n、CM_n、CW_n、CL_n的Kirchhoff指数为

$\begin{array}{c} \mathrm{Kf}\left(\mathrm{CO}_{n}\right)=20 n^{3}+56 n^{2}-34 n ,\\ \mathrm{Kf}\left(\mathrm{CM}_{n}\right)=\frac{80}{3} n^{3}+36 n^{2}-\frac{62}{3} n ,\\ \mathrm{Kf}\left(\mathrm{CW}_{n}\right)=\frac{92}{3} n^{3}+24 n^{2}-\frac{38}{3} n ,\\ \mathrm{Kf}\left(\mathrm{CL}_{n}\right)=32 n^{3}+20 n^{2}-10 n. \end{array}$

下面给出环辛烷链的Kirchhoff指数达到上、下界的极图刻画.

定理2 在所有具有n (n≥3)个正八边形的环辛烷链中, CL_n是具有最大Kirchhoff指数的惟一的环辛烷链, CM_n是具有最小Kirchhoff指数的惟一的环辛烷链.

证明设COC(n; p₁, p₂, p₃)是具有n (n≥3)个正八边形的随机环辛烷链. 由定理1, 有

$\begin{array}{l} f\left(p_{1}, p_{2}, p_{3}\right)=E\left[\operatorname{Kf}\left(\operatorname{COC}\left(n ; p_{1}, p_{2}, p_{3}\right)\right)\right]= \\ \quad \left(-12 n^{3}+36 n^{2}-24 n\right) p_{1}+\left(-\frac{16}{3} n^{3}+16 n^{2}-\frac{32}{3} n\right) p_{2}+ \\ \quad \left(-\frac{4}{3} n^{3}+4 n^{2}-\frac{8}{3} n\right) p_{3}+32 n^{3}+20 n^{2}-10 n . \end{array}$

当n≥3时, 有

$\begin{array}{c} \frac{\partial f}{\partial p_{1}}=-12 n^{3}+36 n^{2}-24 n<0 ,\\ \frac{\partial f}{\partial p_{2}}=-\frac{16}{3} n^{3}+16 n^{2}-\frac{32}{3} n<0 ,\\ \frac{\partial f}{\partial p_{3}}=-\frac{4}{3} n^{3}+4 n^{2}-\frac{8}{3} n<0. \end{array}$

而0≤p₁, p₂, p₃≤1, 0≤p₁+p₂+p₃≤1, 所以f(p₁, p₂, p₃)≤32n³+20n²-10n, 等号成立当且仅当p₁=p₂=p₃=0, 即CL_n是具有最大Kirchhoff指数的惟一的环辛烷链.

另一方面, 有

$\begin{array}{l} f\left(p_{1}, p_{2}, p_{3}\right)=\left(-12 n^{3}+36 n^{2}-24 n\right)\left(p_{1}+p_{2}\right)+ \\ \quad \left(\frac{20}{3} n^{3}-20 n^{2}+\frac{40}{3} n\right) p_{2}+\left(-\frac{4}{3} n^{3}+4 n^{2}-\frac{8}{3} n\right) p_{3}+ \\ \quad 32 n^{3}+20 n^{2}-10 n \geqslant\left(\frac{20}{3} n^{3}-20 n^{2}+\frac{40}{3} n\right) p_{2}+ \\ \quad \left(-\frac{4}{3} n^{3}+4 n^{2}-\frac{8}{3} n\right) p_{3}+20 n^{3}+56 n^{2}-34 n= \\ \quad \left(-\frac{4}{3} n^{3}+4 n^{2}-\frac{8}{3} n\right)\left(p_{2}+p_{3}\right)+\left(\frac{24}{3} n^{3}-24 n^{2}+16 n\right) p_{2}+ \\ \quad 20 n^{3}+56 n^{2}-34 n \geqslant\left(\frac{24}{3} n^{3}-24 n^{2}+16 n\right) p_{2}+\frac{56}{3} n^{3}+ \\ \quad 60 n^{2}-\frac{110}{3} n . \end{array}$

显见

$f\left(p_{1}, p_{2}, p_{3}\right) \geqslant\left(\frac{24}{3} n^{3}-24 n^{2}+16 n\right) p_{2}+\frac{56}{3} n^{3}+60 n^{2}-\frac{110}{3} n.$

(8)

的等号成立当且仅当p₁+p₂=1, p₂+p₃=1且0≤p₁+p₂+p₃≤1, 即(p₁, p₂, p₃)=(0, 1, 0). 将p₂=1代入式(8), 可得f(p₁, p₂, p₃)≥ $\frac{{80}}{3}$ n³+36n²- $\frac{{62}}{3}$ n, 等号成立当且仅当(p₁, p₂, p₃)=(0, 1, 0), 即CM_n是具有最小Kirchhoff指数的惟一的环辛烷链. 证毕.

3. 随机环辛烷链的度积Kirchhoff指数

在有n (n≥2)个正八边形的环辛烷链COC_n中: 第1个和第n个正八边形中恰有1个3度点, 其余7个点都是2度点; 在其余的n-2个正八边形中, 恰有2个3度点, 其余都是2度点; 在O_n中, d(z₁)=3, d(z_i)=2 (2≤i≤8). 所以

$\sum\limits_{v \in V\left(\mathrm{COC}_{n-1}\right)} d_{\mathrm{COC}_{n}}(v)=6(n-1) \times 2+2(n-1) \times 3-1=18 n-19$

且在O_n中,

$\begin{array}{l} \sum\limits_{j=1}^{8} d\left(z_{j}\right) r\left(z_{j}, z_{i}\right)=2 \times \sum\limits_{j=1}^{8} r\left(z_{j}, z_{i}\right)+r\left(z_{1}, z_{i}\right)= \\ \quad \left\{\begin{array}{l} 21\quad (i=1), \\ 21+\frac{7}{8}=\frac{175}{8} \quad(i=2,8), \\ 21+\frac{3}{2}=\frac{45}{2} \quad(i=3,7), \\ 21+\frac{15}{8}=\frac{183}{8} \quad(i=4,6), \\ 21+2=23 \quad(i=5) . \end{array}\right. \end{array}$

(9)

下面求Kf^* (COC(n; p₁, p₂, p₃))的期望值. 记V=V(COC_n), V₁=V(COC_n-1), V₂=V(COC_n-1)\{u_n-1}.

定理3 设COC(n; p₁, p₂, p₃)为具有n (n≥1)个正八边形的随机环辛烷链, 则

$\begin{array}{l} E\left[\mathrm{Kf}^{*}\left(\operatorname{COC}\left(n ; p_{1}, p_{2}, p_{3}\right)\right)\right]= \\ \quad 6\left(27-\frac{81}{8} p_{1}-\frac{9}{2} p_{2}-\frac{9}{8} p_{3}\right) n^{3}+ \\ \quad 18\left(2+\frac{81}{8} p_{1}+\frac{9}{2} p_{2}+\frac{9}{8} p_{3}\right) n^{2}+ \\ \quad 12\left(-\frac{29}{12}-\frac{81}{8} p_{1}-\frac{9}{2} p_{2}-\frac{9}{8} p_{3}\right) n-1 . \end{array}$

(10)

证明令Kf^*(COC_n)=A₁+B₁+C₁, 其中

$\begin{array}{c} A_{1} =\sum\limits_{\{u, v\} \subseteq V_{1}} d(u) d(v) r(u, v), \\ B_{1}=\sum\limits_{v \in V_{1}} \sum\limits_{z_{i} \in V\left(O_{n}\right)} d(v) d\left(z_{i}\right) r\left(v, z_{i}\right) ,\\ C_{1}= \sum\limits_{\left\{z_{i}, z_{j}\right] \subseteq V\left(O_{n}\right)} d\left(z_{i}\right) d\left(z_{j}\right) r\left(z_{i}, z_{j}\right). \end{array}$

由式(2)、(9)及对称性, 类似地, 有

$\begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {A_1} = \sum\limits_{\{ u,v\} \subseteq {V_2}} d (u)d(v)r(u,v) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{v \in {V_2}} {{d_{{\rm{CO}}{{\rm{C}}_n}}}} \left( {{u_{n - 1}}} \right)d(v)r\left( {{u_{n - 1}},v} \right) = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{K}}{{\rm{f}}^*}\left( {{\rm{CO}}{{\rm{C}}_{n - 1}}} \right) + \sum\limits_{v \in {V_1}} d (v)r\left( {{u_{n - 1}},v} \right),\\ {B_1} = \sum\limits_{v \in {V_1}} d (v)\left[ {3\left( {r\left( {v,{u_{n - 1}}} \right) + 1} \right) + 4\left( {r\left( {v,{u_{n - 1}}} \right) + \frac{{15}}{8}} \right) + } \right.\\ \begin{array}{*{20}{l}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 4\left( {r\left( {v,{u_{n - 1}}} \right) + \frac{5}{2}} \right) + 4\left( {r\left( {v,{u_{n - 1}}} \right) + \frac{{23}}{8}} \right) + }\\ {\left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2\left( {r\left( {v,{u_{n - 1}}} \right) + 3} \right)} \right] = 17\sum\limits_{v \in {V_1}} d (v)r\left( {v,{u_{n - 1}}} \right) + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 38(18n - 19),}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {C_1} = \frac{1}{2}\sum\limits_{i = 1}^8 d \left( {{z_i}} \right)\left( {\sum\limits_{j = 1}^8 d \left( {{z_j}} \right)r\left( {{z_j},{z_i}} \right)} \right) = 189.} \end{array} \end{array}$

因此,

$\begin{array}{l} \mathrm{Kf}^{*}\left(\mathrm{COC}_{n}\right)=\mathrm{Kf}^{*}\left(\mathrm{COC}_{n-1}\right)+18 \sum\limits_{v \in V_{1}} d(v) r\left(u_{n-1}, v\right)+ \\ \ \ \ \ \ \ \ \ 684 n-533 . \end{array}$

(11)

而U_n¹=E[ $\sum\limits_{v \in V\left( {C\left( {n;{p_1}, {p_2}, {p_3}} \right)} \right)} {d\left( v \right)r\left( {{u_n}, v} \right)}$ ]. 与U_n类似, 由式(2)、(9)知

$\begin{array}{l} U_{n}^{1}=p_{1} \sum\limits_{v \in V} d(v) r\left(z_{2}, v\right)+p_{2} \sum\limits_{v \in V} d(v) r\left(z_{3}, v\right)+ \\ \quad p_{3} \sum\limits_{v \in V} d(v) r\left(z_{4}, v\right)+\left(1-p_{1}-p_{2}-p_{3}\right) \sum\limits_{v \in V} d(v) r\left(z_{5}, v\right)= \\ \quad p_{1}\left[\sum\limits_{v \in V_{1}} d(v) r\left(u_{n-1}, v\right)+\frac{135}{4} n-\frac{55}{4}\right]+ \\ \quad p_{2}\left[\sum\limits_{v \in V_{1}} d(v) r\left(u_{n-1}, v\right)+45 n-25\right]+ \\ \quad p_{3}\left[\sum\limits_{v \in V_{1}} d(v) r\left(u_{n-1}, v\right)+\frac{207}{4} n-\frac{127}{4}\right]+ \\ \quad \left(1-p_{1}-p_{2}-p_{3}\right)\left[\sum\limits_{v \in V_{1}} d(v) r\left(u_{n-1}, v\right)+54 n-34\right]=U_{n-1}^{1}+ \\ \quad \left(54-\frac{81}{4} p_{1}-9 p_{2}-\frac{9}{4} p_{3}\right) n+\left(-34+\frac{81}{4} p_{1}+9 p_{2}+\frac{9}{4} p_{3}\right). \end{array}$

导入初始值U₁¹= $\sum\limits_{v \in V\left( {{\rm{CO}}{{\rm{C}}_1}} \right)} {d\left( v \right)r\left( {{u_1}, v} \right)}$ =21, 递归可得

$\begin{aligned} U_{n}^{1}=&\left(27-\frac{81}{8} p_{1}-\frac{9}{2} p_{2}-\frac{9}{8} p_{3}\right) n^{2}+\\ &\left(-7+\frac{81}{8} p_{1}+\frac{9}{2} p_{2}+\frac{9}{8} p_{3}\right) n+1. \end{aligned}$

(12)

再由式(11)、(12)可得

$\begin{array}{l} E\left[\operatorname{Kf}^{*}\left(\operatorname{COC}\left(n+1 ; p_{1}, p_{2}, p_{3}\right)\right)\right]= \\ \ \ \ \ E\left[\mathrm{Kf}^{*}\left(\operatorname{COC}\left(n ; p_{1}, p_{2}, p_{3}\right)\right)\right]+18\left(27-\frac{81}{8} p_{1}-\frac{9}{2} p_{2}-\right. \\ \ \ \ \ \left.\frac{9}{8} p_{3}\right) n^{2}+18\left(31+\frac{81}{8} p_{1}+\frac{9}{2} p_{2}+\frac{9}{8} p_{3}\right) n+169, \end{array}$

导入初始值E[Kf^* (COC(1;p₁, p₂, p₃))]=168, 递归可得式(10)成立. 证毕.

将(p₁, p₂, p₃)=(1, 0, 0), (0, 1, 0), (0, 0, 1), (0, 0, 0)代入式(10), 易得下面的推论.

推论2 4种特殊环辛烷链CO_n, CM_n, CW_n, CL_n的度积Kirchhoff指数为

$\begin{array}{c} {\rm{K}}{{\rm{f}}^*}\left( {{\rm{C}}{{\rm{O}}_n}} \right) = \frac{{405}}{4}{n^3} + \frac{{873}}{4}{n^2} - \frac{{301}}{2}n - 1{\rm{ , }}\\ {\rm{K}}{{\rm{f}}^*}\left( {{\rm{C}}{{\rm{M}}_n}} \right) = 135{n^3} + 117{n^2} - 83n - 1,\\ {\rm{K}}{{\rm{f}}^*}\left( {{\rm{C}}{{\rm{W}}_n}} \right) = \frac{{621}}{4}{n^3} + \frac{{225}}{4}{n^2} - \frac{{85}}{2}n - 1,\\ {\rm{K}}{{\rm{f}}^*}\left( {{\rm{C}}{{\rm{L}}_n}} \right) = 162{n^3} + 36{n^2} - 29n - 1. \end{array}$

下面给出环辛烷链的度积Kirchhoff指数达到上、下界的极图刻画.

定理4 在所有具有n (n≥3)个正八边形的环辛烷链中, CL_n是具有最大度积Kirchhoff指数的惟一的环辛烷链, CM_n是具有最小度积Kirchhoff指数的惟一的环辛烷链.

证明设COC(n; p₁, p₂, p₃)为具有n (n≥3)个正八边形的随机环辛烷链. 由定理3, 有

$\begin{array}{l} f_{1}\left(p_{1}, p_{2}, p_{3}\right)=E\left[\mathrm{Kf}^{*}\left(\operatorname{COC}\left(n ; p_{1}, p_{2}, p_{3}\right)\right)\right]= \\ \quad \begin{array}{l} \left(-\frac{243}{4} n^{3}+\frac{729}{4} n^{2}-\frac{243}{2} n\right) p_{1}+\left(-27 n^{3}+81 n^{2}-54 n\right) p_{2}+ \\ \left(-\frac{27}{4} n^{3}+\frac{81}{4} n^{2}-\frac{27}{2} n\right) p_{3}+162 n^{3}+36 n^{2}-29 n-1. \end{array} \end{array}$

当n≥3时, 有

$\begin{array}{c} \frac{\partial f_{1}}{\partial p_{1}}=-\frac{243}{4} n^{3}+\frac{729}{4} n^{2}-\frac{243}{2} n<0 \\ \frac{\partial f_{1}}{\partial p_{2}}=-27 n^{3}+81 n^{2}-54 n<0 ,\\ \frac{\partial f_{1}}{\partial p_{3}}=-\frac{27}{4} n^{3}+\frac{81}{4} n^{2}-\frac{27}{2} n<0. \end{array}$

而0≤p₁, p₂, p₃≤1, 0≤p₁+p₂+p₃≤1, 所以f₁(p₁, p₂, p₃)≤162n³+36n²-29n-1, 等号成立当且仅当p₁=p₂=p₃=0, 即CL_n是具有最大度积Kirchhoff指数的惟一的环辛烷链.

另一方面, 有

$\begin{array}{l} f_{1}\left(p_{1}, p_{2}, p_{3}\right)=\left(-\frac{243}{4} n^{3}+\frac{729}{4} n^{2}-\frac{243}{2} n\right)\left(p_{1}+p_{2}\right)+ \\ \quad \left(\frac{135}{4} n^{3}-\frac{405}{4} n^{2}+\frac{135}{2} n\right) p_{2}+\left(-\frac{27}{4} n^{3}+\frac{81}{4} n^{2}-\frac{27}{2} n\right) p_{3}+ \\ \quad 162 n^{3}+36 n^{2}-29 n-1 \geqslant\left(\frac{135}{4} n^{3}-\frac{405}{4} n^{2}+\frac{135}{2} n\right) p_{2}+ \\ \quad \left(-\frac{27}{4} n^{3}+\frac{81}{4} n^{2}-\frac{27}{2} n\right) p_{3}+\frac{405}{4} n^{3}+\frac{873}{4} n^{2}-\frac{301}{2} n-1= \\ \quad \left(-\frac{27}{4} n^{3}+\frac{81}{4} n^{2}-\frac{27}{2} n\right)\left(p_{2}+p_{3}\right)+\left(\frac{81}{2} n^{3}-\frac{243}{2} n^{2}+81 n\right) p_{2}+ \\ \quad \frac{405}{4} n^{3}+\frac{873}{4} n^{2}-\frac{301}{2} n-1 \geqslant\left(\frac{81}{2} n^{3}-\frac{243}{2} n^{2}+81 n\right) p_{2}+ \\ \quad \frac{189}{2} n^{3}+\frac{477}{2} n^{2}-164 n-1. \end{array}$

显见

$\begin{array}{l} f_{1}\left(p_{1}, p_{2}, p_{3}\right) \geqslant\left(\frac{81}{2} n^{3}-\frac{243}{2} n^{2}+81 n\right) p_{2}+\frac{189}{2} n^{3}+ \\ \ \ \ \ \ \ \frac{477}{2} n^{2}-164 n-1 \end{array}$

(13)

的等号成立当且仅当p₁+p₂=1, p₂+p₃=1且0≤p₁+p₂+p₃≤1, 即(p₁, p₂, p₃)=(0, 1, 0). 将p₂=1代入式(13)可得f₁(p₁, p₂, p₃)≥135n³+117n²-83n-1, 等号成立当且仅当(p₁, p₂, p₃)=(0, 1, 0), 即CM_n是具有最小度积Kirchhoff指数的惟一的环辛烷链. 证毕.

4. 随机环辛烷链的度和Kirchhoff指数

本节求Kf⁺(COC(n; p₁, p₂, p₃))的期望值. 仍记V=V(COC_n), V₁=V(COC_n-1), V₂=V(COC_n-1)\{u_n-1}.

定理5 设COC(n; p₁, p₂, p₃)为具有n (n≥1)个正八边形的随机环辛烷链, 则

$\begin{array}{r} E\left[\mathrm{Kf}^{+}\left(\operatorname{COC}\left(n ; p_{1}, p_{2}, p_{3}\right)\right)\right]=6\left(24-9 p_{1}-4 p_{2}-p_{3}\right) n^{3}+ \\ 18\left(\frac{61}{18}+9 p_{1}+4 p_{2}+p_{3}\right) n^{2}-12\left(\frac{37}{12}+9 p_{1}+4 p_{2}+p_{3}\right) n. \end{array}$

(14)

证明令Kf⁺(COC_n)=A₂+B₂+C₂, 其中, A₂= $\sum\limits_{\left\{ {u, v} \right\} \subseteq {V_1}} {\left( {d\left( u \right) + d\left( v \right)} \right)r\left( {u, v} \right)}$ , B₂= $\sum\limits_{v \in {V_1}} {\sum\limits_{{z_i} \in V\left( {{O_n}} \right)} {\left( {d\left( v \right) + d\left( {{z_i}} \right)} \right)r\left( {v, {z_i}} \right)} }$ , C₂= $\sum\limits_{\left\{ {{z_i}, {z_j}} \right\} \subseteq V\left( {{O_n}} \right)} {\left( {d\left( {{z_i}} \right) + d\left( {{z_j}} \right)} \right)r\left( {{z_i}, {z_j}} \right)}$ .

由式(2)、(9), 类似地, 有

$\begin{array}{l} \quad A_{2}=\sum\limits_{\{u, v\} \subseteq V_{2}}(d(u)+d(v)) r(u, v)+ \\ \quad \quad \quad \sum\limits_{v \in V_{2}}\left(d_{\mathrm{Coc}_{n}}\left(u_{n-1}\right)+d(v)\right) r\left(u_{n-1}, v\right)= \\ \quad \quad \quad \mathrm{Kf}^{+}\left(\mathrm{COC}_{n-1}\right)+r\left(u_{n-1} \mid \mathrm{COC}_{n-1}\right) \\ B_{2}=\sum\limits_{v \in V_{1}} \sum\limits_{z_{i} \in V\left(\sigma_{n}\right)} d(v) r\left(v, z_{i}\right)+\quad \sum\limits_{v \in V_{1}} \sum\limits_{z_{i} \in V\left(O_{n}\right)} d\left(z_{i}\right) r\left(v, z_{i}\right)= \\ \quad \quad 8 \sum\limits_{v \in V_{1}} d(v) r\left(v, u_{n-1}\right)+17 r\left(u_{n-1} \mid \mathrm{COC}_{n-1}\right)+ \\ \quad \quad \frac{37}{2}(18 n-19)+304(n-1), \\ \quad \quad C_{2}=\frac{1}{2} \sum\limits_{i=1}^{8} \sum\limits_{j=1}^{8}\left(d\left(z_{i}\right)+d\left(z_{j}\right)\right) r\left(z_{i}, z_{j}\right)=\frac{357}{2} . \end{array}$

因此,

$\begin{array}{l} \mathrm{Kf}^{+}\left(\mathrm{COC}_{n}\right)=\mathrm{Kf}^{+}\left(\mathrm{COC}_{n-1}\right)+18 r\left(u_{n-1} \mid \mathrm{COC}_{n-1}\right)+ \\ \quad 8 \sum\limits_{v \in V_{1}} d(v) r\left(u_{n-1}, v\right)+637 n-477 . \end{array}$

(15)

记U_n²=E[r(u_n|COC(n; p₁, p₂, p₃))], 与U_n类似, 由式(2)、(3)知

$\begin{aligned} U_{n}^{2}=& r\left(z_{2} \mid \mathrm{COC}_{n}\right) p_{1}+r\left(z_{3} \mid \mathrm{COC}_{n}\right) p_{2}+r\left(z_{4} \mid \mathrm{COC}_{n}\right) p_{3}+\\ & r\left(z_{5} \mid \mathrm{COC}_{n}\right)\left(1-p_{1}-p_{2}-p_{3}\right)=\\ & U_{n-1}^{2}+\left(24-9 p_{1}-4 p_{2}-p_{3}\right) n+\left(-\frac{27}{2}+9 p_{1}+4 p_{2}+p_{3}\right). \end{aligned}$

代入初始值U₁²=21/2, 递归可得

$U_{n}^{2}=\frac{1}{2}\left(24-9 p_{1}-4 p_{2}-p_{3}\right) n^{2}+\frac{1}{2}\left(-3+9 p_{1}+4 p_{2}+p_{3}\right) n .$

(16)

由式(15)、(16), 可得

$\begin{array}{l} E\left[\mathrm{Kf}^{+}\left(\operatorname{COC}\left(n+1 ; p_{1}, p_{2}, p_{3}\right)\right)\right]=\\ \ \ \ \ E\left[\mathrm{Kf}^{+}\left(\operatorname{COC}\left(n ; p_{1}, p_{2}, p_{3}\right)\right)\right]+18\left(24-9 p_{1}-4 p_{2}-p_{3}\right) n^{2}+\\ \ \ \ \ 18\left(\frac{277}{9}+9 p_{1}+4 p_{2}+p_{3}\right) n+168 . \end{array}$

代入初始值E[Kf⁺(COC(1;p₁, p₂, p₃))]=168, 递归可得式(14)成立. 证毕.

将(p₁, p₂, p₃)=(1, 0, 0), (0, 1, 0), (0, 0, 1), (0, 0, 0)分别代入式(14), 易得如下推论.

推论3 4种特殊环辛烷链CO_n, CM_n, CW_n, CL_n的度和Kirchhoff指数为

$\begin{array}{c} \mathrm{Kf}^{+}\left(\mathrm{CO}_{n}\right)=90 n^{3}+223 n^{2}-145 n, \\ \mathrm{Kf}^{+}\left(\mathrm{CM}_{n}\right)=120 n^{3}+133 n^{2}-85 n, \\ \mathrm{Kf}^{+}\left(\mathrm{CW}_{n}\right)=138 n^{3}+79 n^{2}-49 n, \\ \mathrm{Kf}^{+}\left(\mathrm{CL}_{n}\right)=144 n^{3}+61 n^{2}-37 n. \end{array}$

下面给出环辛烷链的度和Kirchhoff指数达到上、下界的极图刻画.

定理6 在所有具有n (n≥3)个正八边形的环辛烷链中, CL_n是具有最大度和Kirchhoff指数的惟一的环辛烷链, CM_n是具有最小度和Kirchhoff指数的惟一的环辛烷链.

证明设COC(n; p₁, p₂, p₃)为具有n (n≥3)个正八边形的随机环辛烷链. 由定理5, 有

$\begin{array}{l} f_{2}\left(p_{1}, p_{2}, p_{3}\right)=E\left[\mathrm{Kf}^{+}\left(\operatorname{COC}\left(n ; p_{1}, p_{2}, p_{3}\right)\right)\right]= \\ \ \ \ \ \left(-54 n^{3}+162 n^{2}-108 n\right) p_{1}+\left(-24 n^{3}+72 n^{2}-48 n\right) p_{2}+ \\ \ \ \ \ \left(-6 n^{3}+18 n^{2}-12 n\right) p_{3}+144 n^{3}+61 n^{2}-37 n. \end{array}$

当n≥3时, 有

$\begin{array}{c} \frac{\partial f_{2}}{\partial p_{1}}=-54 n^{3}+162 n^{2}-108 n<0, \\ \frac{\partial f_{2}}{\partial p_{2}}=-24 n^{3}+72 n^{2}-48 n<0 ,\\ \frac{\partial f_{2}}{\partial p_{3}}=-6 n^{3}+18 n^{2}-12 n<0 . \end{array}$

而0≤p₁, p₂, p₃≤1, 0≤p₁+p₂+p₃≤1, 所以f₂(p₁, p₂, p₃)≤144n³+61n²-37n, 等号成立当且仅当p₁=p₂=p₃=0, 即CL_n是具有最大度和Kirchhoff指数的惟一的环辛烷链.

另一方面, 有

$\begin{array}{l} f_{2}\left(p_{1}, p_{2}, p_{3}\right)=\left(-54 n^{3}+162 n^{2}-108 n\right)\left(p_{1}+p_{2}\right)+ \\ \ \ \ \ \left(30 n^{3}-90 n^{2}+60 n\right) p_{2}+\left(-6 n^{3}+18 n^{2}-12 n\right) p_{3}+ \\ \ \ \ \ 162 n^{3}+144 n^{3}+61 n^{2}-37 n \geqslant\left(30 n^{3}-90 n^{2}+60 n\right) p_{2}+ \\ \ \ \ \ \left(-6 n^{3}+18 n^{2}-12 n\right) p_{3}+90 n^{3}+223 n^{2}-145 n= \\ \ \ \ \ \left(-6 n^{3}+18 n^{2}-12 n\right)\left(p_{2}+p_{3}\right)+\left(36 n^{3}-108 n^{2}+72 n\right) p_{2}+ \\ \ \ \ \ 90 n^{3}+223 n^{2}-145 n \geqslant\left(36 n^{3}-108 n^{2}+72 n\right) p_{2}+84 n^{3}+ \\ \ \ \ \ 241 n^{2}-157 n . \end{array}$

显见

$\begin{array}{l} f_{2}\left(p_{1}, p_{2}, p_{3}\right) \geqslant\left(36 n^{3}-108 n^{2}+72 n\right) p_{2}+ \\ \ \ \ \ 84 n^{3}+241 n^{2}-157 n. \end{array}$

(17)

的等号成立当且仅当p₁+p₂=1, p₂+p₃=1且0≤p₁+p₂+p₃≤1, 即(p₁, p₂, p₃)=(0, 1, 0). 将p₂=1代入式(17)，可得f₂(p₁, p₂, p₃)≥120n³+133n²-85n, 等号成立当且仅当(p₁, p₂, p₃)=(0, 1, 0), 即CM_n是具有最小度和Kirchhoff指数的惟一的环辛烷链. 证毕.

5. 随机环辛烷链的3类Kirchhoff指数的平均值

记 $\mathcal{H}$ _n为所有具有n个正八边形的环辛烷链的集合, 则 $\mathcal{H}$ _n中所有环辛烷链的3类Kirchhoff指数的平均值记为 $\mathcal{H}$ _n的3类Kirchhoff指数的平均值, 其表达式分别为

$\begin{array}{c} \mathrm{Kf}_{\mathrm{avr}}\left(\mathcal{H}_{n}\right)=\frac{1}{\left|\mathcal{H}_{n}\right|} \sum\limits_{G \in \mathcal{H}_{n}} \mathrm{Kf}(G) ,\\ \mathrm{Kf}_{\mathrm{avr}}^{*}\left(\mathcal{H}_{n}\right)=\frac{1}{\left|\mathcal{H}_{n}\right|} \sum\limits_{G \in \mathcal{H}_{n}} \mathrm{Kf}^{*}(G) ,\\ \mathrm{Kf}_{\mathrm{avr}}^{+}\left(\mathcal{H}_{n}\right)=\frac{1}{\left|\mathcal{H}_{n}\right|} \sum\limits_{G \in \mathcal{H}_{n}} \mathrm{Kf}^{+}(G). \end{array}$

由于 $\mathcal{H}$ _n中每条随机链的产生都是等可能的, 故而当(p₁, p₂, p₃, 1-p₁-p₂-p₃)=(1/4, 1/4, 1/4, 1/4)时, 可以得到其平均值. 将(p₁, p₂, p₃)=(1/4, 1/4, 1/4)代入定理1、定理3、定理5的期望表达式, 易得下面的结论.

定理7 环辛烷链集 $\mathcal{H}$ _n的3类Kirchhoff指数的平均值分别为

$\begin{array}{c} \mathrm{Kf}_{\text {avr }}\left(\mathcal{H}_{n}\right)=\frac{82}{3} n^{3}+34 n^{2}-\frac{58}{3} n ,\\ \mathrm{Kf}_{\text {avr }}^{*}\left(\mathcal{H}_{n}\right)=\frac{1107}{8} n^{3}+\frac{855}{8} n^{2}-\frac{305}{4} n-1, \\ \mathrm{Kf}_{\text {avr }}^{+}\left(\mathcal{H}_{n}\right)=123 n^{3}+124 n^{2}-79 n. \end{array}$

我们发现

$\begin{array}{l} \mathrm{K} \mathrm{f}_{\mathrm{av}}\left(\mathcal{H}_{n}\right)=\frac{1}{4}\left(\mathrm{Kf}\left(\mathrm{CO}_{n}\right)+\mathrm{Kf}\left(\mathrm{CM}_{n}\right)+\mathrm{Kf}\left(\mathrm{CW}_{n}\right)+\mathrm{Kf}\left(\mathrm{CL}_{n}\right)\right),\\ \ \ \ \ \ \ \ \ \mathrm{Kf}_{\text {avr }}^{*}\left(\mathcal{H}_{n}\right)=\frac{1}{4}\left(\mathrm{Kf}^{*}\left(\mathrm{CO}_{n}\right)+\mathrm{Kf}^{*}\left(\mathrm{CM}_{n}\right)+\right. \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left.\mathrm{Kf}^{*}\left(\mathrm{CW}_{n}\right)+\mathrm{Kf}^{*}\left(\mathrm{CL}_{n}\right)\right), \\ \mathrm{Kf}_{\text {avr }}^{+}\left(\mathcal{H}_{n}\right)=\frac{1}{4}\left(\mathrm{Kf}^{+}\left(\mathrm{CO}_{n}\right)+\mathrm{Kf}^{+}\left(\mathrm{CM}_{n}\right)+\mathrm{Kf}^{+}\left(\mathrm{CW}_{n}\right)+\right. \\ \ \ \ \ \ \ \ \ \left.\mathrm{Kf}^{+}\left(\mathrm{CL}_{n}\right)\right). \end{array}$

这意味着可以用这4条特殊的链CO_n、CM_n、CW_n、CL_n的Kirchhoff指数(度积Kirchhoff指数、度和Kirchhoff指数)的平均值来表示 $\mathcal{H}$ _n的Kirchhoff指数(度积Kirchhoff指数、度和Kirchhoff指数)整体的平均值.

图 1 n个正八边形的环辛烷链COC_n

Figure 1. A cyclooctane chain with n regular octagons

下载: 全尺寸图片幻灯片

图 2 环辛烷链的4种连接方式

Figure 2. Four connection methods of cyclooctane chains

下载: 全尺寸图片幻灯片

参考文献(24)

[1]	WIENER H. Structural determination of paraffin boiling points[J]. Journal of the American Chemical Society, 1947, 69(1): 17-20. doi: 10.1021/ja01193a005
[2]	HOSOYA H. Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons[J]. Bulletin of the Chemi-cal Society of Japan, 1971, 44(9): 2332-2339. doi: 10.1246/bcsj.44.2332
[3]	PLAVŠI Ć D, NIKOLI Ć S, TRINAJSTI Ć N, et al. On the Harary index for the characterization of chemical graphs[J]. Journal of Mathematical Chemistry, 1993, 12: 235-250. doi: 10.1007/BF01164638
[4]	BONCHEV D, BALABAN A T, LIU X Y, et al. Molecular cyclicity and centricity of polycyclic graphs. I. Cyclicity based on resistance distances or reciprocal distances[J]. International Journal of Quantum Chemistry, 1994, 50(1): 1-20. doi: 10.1002/qua.560500102
[5]	WU T Z, LU H Z. Hyper-Wiener indices of polyphenyl chains and polyphenyl spiders[J]. Open Mathematics, 2019, 17(1): 668-676. doi: 10.1515/math-2019-0053
[6]	YANG W L, ZHANG F J. Wiener index in random polyphenyl chains[J]. MATCH Communications in Mathematical and in Computer Chemistry, 2012, 68(1): 371-376. http://d.wanfangdata.com.cn/periodical/02a0b21332dbc125b299bf2df835683c
[7]	DENG H Y. Wiener indices of spiro and polyphenyl hexagonal chains[J]. Mathematical and Computer Modelling, 2012, 55(3/4): 634-644. http://d.wanfangdata.com.cn/periodical/02a0b21332dbc125b299bf2df835683c
[8]	DENG H Y, TANG Z K. Kirchhoff indices of spiro and polyphenyl hexagonal chains[J]. Utilitas Mathematica, 2014, 95: 113-128. http://smartsearch.nstl.gov.cn/paper_detail.html?id=762c56610dade26481ee269cf4932071
[9]	WANG H Y, JIANG Q, GUTMAN I. Wiener numbers of random pentagonal chains[J]. Iranian Journal of Mathematical Chemistry, 2013, 4(1): 59-76. http://www.researchgate.net/publication/298092974_Wiener_numbers_of_random_pentagonal_chains
[10]	MA L, BIAN H, LIU B J, et al. The expected values of the Wiener indices in the random phenylene and spiro chains[J]. Ars Combinatoria, 2017, 130: 267-274. http://smartsearch.nstl.gov.cn/paper_detail.html?id=c38e709fd0f9bae694afadab6faa0e81
[11]	HUANG G H, KUANG M J, DENG H Y. The expected values of Kirchhoff indices in the random polyphenyl and spiro chains[J]. Ars Mathematics Contematica, 2014, 9(2): 197-207. doi: 10.26493/1855-3974.458.7b0
[12]	WEI S L, KE X L, WANG Y. Wiener indices in random cyclooctane chains[J]. Wuhan University Journal of Natu-ral Sciences, 2018, 23(6): 498-502. doi: 10.1007/s11859-018-1355-5
[13]	ZHANG L L, LI Q S, LI S C, et al. The expected values for the Schultz index, Gutman index, multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index of a random polyphenylene chain[J]. Discrete Applied Mathematics, 2020, 282: 243-256. doi: 10.1016/j.dam.2019.11.007
[14]	YANG Y J, JIANG X Y. Unicyclic graphs with extremal Kirchhoff index[J]. MATCH Communications in Mathematical and in Computer Chemistry, 2008, 60(1): 107-120. http://www.researchgate.net/publication/254258266_Unicyclic_graphs_with_extremal_Kirchhoff_index
[15]	GUO Q Z, DENG H Y, CHEN D D. The extremal Kirchhoff index of a class of unicyclic graphs[J]. MATCH Communications in Mathematical and in Computer Che-mistry, 2009, 61(3): 713-722. http://www.researchgate.net/publication/266704508_The_extremal_Kirchhoff_index_of_a_class_of_unicyclic_graphs
[16]	FENG L H, LIU W J, YU G H, et al. The degree Kirchhoff index of fully loaded unicyclic graphs and cacti[J]. Utili-tas Mathematica, 2014, 95: 149-159. http://smartsearch.nstl.gov.cn/paper_detail.html?id=169152d648bcb79178c4a57157ac0dbb
[17]	FEI J Q, TU J H. Complete characterization of bicyclic graphs with the maximum and second-maximum degree Kirchhoff index[J]. Applied Mathematics and Computation, 2018, 330: 118-124. doi: 10.1016/j.amc.2018.02.025
[18]	ZHOU B, TRINAJSTI Ć N. On resistance-distance and Kirchhoff index[J]. Journal of Mathematical Chemistry, 2009, 46(1): 283-289. doi: 10.1007/s10910-008-9459-3
[19]	YOU Z F, YOU L H, HONG W X. Comment on "Kirchhoff index in line, subdivision and total graphs of a regular graph"[J]. Discrete Applied Mathematics, 2013, 161(18): 3100-3103. doi: 10.1016/j.dam.2013.06.015
[20]	WANG H Z, HUA H B, WANG D D. Cacti with minimum, second-minimum, and third-minimum Kirchhoff indices[J]. Mathematical Communications, 2010, 15(2): 347-358. http://www.ams.org/mathscinet-getitem?mr=2814296
[21]	TANG Z K, DENG H Y. Degree Kirchhoff index of bicyclic graphs[J]. Canadian Mathematical Bulletin, 2017, 60(1): 197-205. doi: 10.4153/CMB-2016-063-5
[22]	KLEIN D J, RANDI Ć M. Resistance distance[J]. Journal of Mathematical Chemistry, 1993, 12(1): 81-95. doi: 10.1007/BF01164627
[23]	CHEN H Y, ZHANG F J. Resistance distance and the normalized Laplacian spectrum[J]. Discrete Applied Mathematics, 2017, 155: 654-661. http://www.ams.org/mathscinet-getitem?mr=2303977
[24]	GUTMAN I, FENG L H, YU G H. Degree resistance distance of unicyclic graphs[J]. Transactions on Combina-torics, 2012, 1: 27-40. http://www.oalib.com/paper/2304841

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1. 预备知识
2. 随机环辛烷链的Kirchhoff指数
3. 随机环辛烷链的度积Kirchhoff指数
4. 随机环辛烷链的度和Kirchhoff指数
5. 随机环辛烷链的3类Kirchhoff指数的平均值

随机环辛烷链的三类Kirchhoff指数

通讯作者: 尤利华, Email: ylhua@scnu.edu.cn

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