柴文麗，田文文

(1.西北民族大學美術(shù)學院，甘肅蘭州730030；2.西北民族大學 數(shù)學與計算機科學學院，甘肅蘭州730030)

一類三圈圖關(guān)于Merrifield-Simmons指標和Hosoya指標的排序

柴文麗1，田文文2

(1.西北民族大學美術(shù)學院，甘肅蘭州730030；2.西北民族大學 數(shù)學與計算機科學學院，甘肅蘭州730030)

三圈圖；Merrifield-Simmons指標；Hosoya指標；排序

0 引言

1 預備知識

引理1[4]設(shè)G是一個簡單的連通圖，對任意的u，v∈V(G),uv∈E(G),則σ(G)=σ(G-{v})+σ(G-NG[v])；σ(G)=σ(G-{uv})-σ(G-(NG[u]∪NG[v]))．

引理3[4]若G1,G2,…，Gk是圖G的連通分支，則

引理4[4]對于n階的路Pn，有σ(Pn)=fn+2;μ(Pn)=fn+1.

引理5[4]對于n階的圈Cn，有σ(Cn)=fn+1+fn-1;μ(Cn)=fn+1+fn-1.

由引理1～5可得以下結(jié)論：

引理7 對于如圖2所示的圖H，有

1)σ(H)=fm+2fn+2·(fq+1fr+fq-1fr-1)+fm+1fn+1·(fq+1fr-1+fq-1fr-2)；

2)μ(H)= (fq+1+fq-1)(fm+1fn+1fr+fm+1fnfr-1+fmfn+1fr-1)

+fq·(fm+1fn+1fr-1+fm+1fnfn-2+fmfn+1fr-2).

2 主要結(jié)果及其證明

證明 如圖1所示，由引理1及引理7可知

(fq+1fr-1+fq-1fr-2)(fm+1fkfp-k+fm-1fk-1fp-k-1).

(fq+1fr-1+fq-1fr-2)(fm+1fk+1fp-k-1+fm-1fkfp-k-2).

所以，由引理6可知

-fkfp-k)]+(fq+1fr-1+fq-1fr-2)[fm+1·(fk+1fp-k-1-fkfp-k)+fm-1·(fkfp-k-2-fk-1fp-k-1)]

+(fq+1fr-1+fq-1fr-2)[fm+1·(lp-2k+lp-2k-2)+fm+1·(-lp-2k-lp-2k-2)]}

而fm·(-fq+1fr-2-fq-1fr-3)<0.

+(fq+1fr-1+fq-1fr-2)(fm+1fk+2fp-k-2+fm-1fk+1fp-k-3),

所以由引理6可知

-fkfp-k)]+(fq+1fr-1+fq-1fr-2)[fm+1·(fk+2fp-k-2-fkfp-k)+fm-1·(fk+1fp-k-3-fk-1fp-k-1)]

+(fq+1fr-1+fq-1fr-2)[fm+1·(lp-2k-lp-2k-4)+fm+1·(lp-2k-4-lp-2k)]}

而fm·(lp-2k-4-lp-2k)(fq+1fr-2+fq-1fr-3)<0，

證明 由引理2及引理7可知

+fq·[(fm+1+fm-1)(fkfp-kfr-1+fkfp-k-1fr-2+fk-1fp-kfr-2)+fm(fkfp-k-1fr-1+

fkfp-k-2fr-2+2fk-1fp-k-1fr-2+fk-1fp-kfr-1+fk-2fp-kfr-2)].

+fm·(fk+1fp-k-2fr+fk+1fp-k-3fr-1+2fkfp-k-2fr-1+fkfp-k-1fr+fk-1fp-k-1fr-1)]

fq·[(fm+1+fm-1)(fk+1fp-k-1fr-1+fk+1fp-k-2fr-2+fkfp-k-1fr-2)+fm·

(fk+1fp-k-2fr-1+fk+1fp-k-3fr-2+2fkfp-k-2fr-2+fkfp-k-1fr-1+fk-1fp-k-1fr-2)].

所以，由引理6可知

-fk-1fp-kfr-1)+fm·(fk+1fp-k-2fr+fk+1fp-k-3fr-1+fkfp-k-2fr-1-fk-1fp-k-1fr-1-fk-1fp-kfr

-fk-2fp-kfr-1)]+fq·[(fm+1+fm-1)(fk+1fp-k-1fr-1+fk+1fp-k-2fr-2-fkfp-kfr-1-fk-1fp-kfr-2)

由表1可知，插秧機插植部兩側(cè)浮板的最大高度差可以達到6.1cm，插植部工作中期望最大傾角達到5.71°。在靜態(tài)試驗中秧苗插深自適應(yīng)調(diào)節(jié)系統(tǒng)的控制相對誤差均在5%以內(nèi)，其中當該系統(tǒng)工作在半量程區(qū)間時，有著較好的控制精度。而系統(tǒng)處于初始以及接近滿量程工作區(qū)間時，控制精度較差，相對誤差在4%以上。究其原因，這應(yīng)是該系統(tǒng)橫向仿形控制機構(gòu)的機械結(jié)構(gòu)設(shè)計引起的。

+fm·(fk+1fp-k-2fr-1+fk+1fp-k-3fr-2+fkfp-k-2fr-2-fk-1fp-k-1fr-2-fk-1fp-kfr-1-fk-2fp-kfr-2)]

[fr-1·(fq+1+fq-1)(fm+1+fm-1)+fr-2fm·(fq+1+fq-1)+fr-2fq·(fm+1+fm-1)+fr-3fqfm]}.

又因(lp-2k+lp-2k+2)-(lp-2k+lp-2k-2)>0,

(fm+1+fm-1)(fq+1fr+fq-1fr+fqfr-1)-[fr-1·(fq+1+fq-1)(fm+1+fm-1)+fr-2fm·(fq+1+fq-1)

+fr-2fq·(fm+1+fm-1)+fr-3fqfm]=2fm-1fr-2·(fq+1+fq-1)+2fm-1fr-3fq>0,

所以(fm+1+fm-1)(fq+1fr+fq-1fr+fqfr-1)(lp-2k+lp-2k+2)-(lp-2k+lp-2k-2)[fr-1·(fq+1+fq-1)

(fm+1+fm-1)+fr-2fm·(fq+1+fq-1)+fr-2fq·(fm+1+fm-1)+fr-3fqfm]>0.

+fm·(fk+2fp-k-3fr+fk+2fp-k-4fr-1+2fk+1fp-k-3fr-1+fk+1fp-k-2fr+fkfp-k-2fr-1)]

+fq·[(fm+1+fm-1)(fk+2fp-k-2fr-1+fk+2fp-k-3fr-2+fk+1fp-k-2fr-2)

+fm·(fk+2fp-k-3fr-1+fk+2fp-k-4fr-2+2fk+1fp-k-3fr-2+fk+1fp-k-2fr-1+fkfp-k-2fr-2)].

所以由引理6可知

(fk+2fp-k-3+fk+1fp-k-2-fkfp-k-1-fk-1fp-k)·fr-1+fmfr·(fk+2fp-k-3+fk+1fp-k-2-fkfp-k-1-

fk-1fp-k)+fmfr-1·(fk+2fp-k-4+2fk+1fp-k-3-2fk-1fp-k-1-fk-2fp-k)]+fq·[(fm+1+fm-1)

(fk+2fp-k-2-fkfp-k)·fr-1+(fm+1+fm-1)(fk+2fp-k-3+fk+1fp-k-2-fkfp-k-1-fk-1fp-k)

·fr-2+fmfr-1·(fk+2fp-k-3+fk+1fp-k-2-fkfp-k-1-fk-1fp-k)+fmfr-2·

(fk+2fp-k-4+2fk+1fp-k-3-2fk-1fp-k-1-fk-2fp-k)]

-fqfr-3fm](lp-2k-1+lp-2k-3)

而2fm-1fr-2·(fq+1+fq-1)+2fqfm-1fr-3>0.

[1] Bondy J A，Murty U S R．Graph theory with applications[M]．New York：The Macmillan Press，1976．

[2] Hosoya H．Topological index[J]．Bull Chem Soc Japan，1971，44：2332-2339．

[3] Merrfield R E，Simmons H E．Topological Methods in Chemistry[M].New York：Wiley，1989．

[4] Gutman I，Polansky O E．Mathematical Concepts in Organic Chemistry[M]．Berlin：Spring-er，1986．

[5] Gutman I，Cyvin S J．Introduction to the Theory of Benzenoid Hydrocarbons[M]．Berlin：Springer，1989．

[6] Deng H Y，Chen S B，Zhang J．The Merrifield-Simmons index in-graphs[J]．Journal of Mathematical Chemistry,2008，43(1)：75-91．

[7] Deng H Y．The smallest Merrifield-Simmons index of -graphs[J]．Math Comput Model,2009，49(1-2)：320-326．

[8] Deng H Y．The smallest Hosoya index in -graphs[J]．Journal of Mathematical Chemistry，2008，43(1)：119-133．

[9] Xu Kexiang，Gutman I．The Greatest Hosoya Index of Bicyclic Graphs with Given Maximum Degree[J]．MATCH Commun Math Comput Chem,2011，66(3)：795-824．

[10] 周旭冉，王力工．一類雙圈圖的兩種指標的排序[J]．山東大學學報(理學版),2011，46(11)：44-47．

[11] Zhu Zhongxun，Li Shuchao，Tan Liansheng．Tricyclic graphs with maximum Merrifield-Simmons index[J]．Discrete Applied Mathematics,2010，158(3)：204-212．

[12] Dolati A，Haghighat M，Golalizadeh S，Safari M．The Smallest Hosoya index of Connected Tricyclic Graphs[J]．MATCH Commun Math Comput Chem,2011，65(1)：57-70．

[13] Zhu Zhongxun，Yu Qigang．The number of independent sets of tricyclic graphs[J]．Applied Mathematics Letters,2012，25(10)：1327-1334．

[14] Xuezheng Lv，Yan Yan，Aimei Yu，Jingjing Zhang．Ordering strees with given pendent vertices with respect to Merrifield-Simmons indices and Hosoya indices[J]．Journal of Mathemati-cal Chemistry,2010，47：11-20．

[15] Stephan G Wagner．Extremal trees with respect to Hosoya index and Merrifield-Simmons index[J]．MATCH Commun Math Comput Chem,2007，57(1)：221-233．

Orderings of a Class of Tricyclic Graphs with Respect to Merrifield-Simmons and Hosoya Indexes

CHAI Wen-li1，TIAN Wen-wen2

(1. School of Fine Arts，Northwest University for Nationalities，Lanzhou 730030，China；2. School of Mathematics and Computer Science，Northwest University for Nationalities，Lanzhou 730030，China)

The Merrifield-Simmons index and Hosoya index of the class of tricyclic graphs were investigated according to the distance between and on.Their orderings with respect to these two indices had been obtained．

Tricyclic graphs；Merrifield-Simmons index；Hosoya index；order

2015-11-20

國家民委科研項目(14XBZ018)；甘肅省自然科學基金(145RJZA158)；西北民族大學科研創(chuàng)新團隊計劃資助項目；中央高?；究蒲袠I(yè)務(wù)費專項資金項目(31920140059)．

柴文麗(1988— )，女，甘肅天水人，碩士研究生.

O157.5

A

1009-2102(2015)04-0001-05