一类三圈图关于Merrifield-Simmons指标和Hosoya指标的排序

柴文丽，田文文

(1.西北民族大学美术学院，甘肃兰州730030；2.西北民族大学 数学与计算机科学学院，甘肃兰州730030)

一类三圈图关于Merrifield-Simmons指标和Hosoya指标的排序

柴文丽1，田文文2

(1.西北民族大学美术学院，甘肃兰州730030；2.西北民族大学 数学与计算机科学学院，甘肃兰州730030)

三圈图；Merrifield-Simmons指标；Hosoya指标；排序

0 引言

1 预备知识

引理1[4]设G是一个简单的连通图，对任意的u，v∈V(G),uv∈E(G),则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是图G的连通分支，则

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

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

由引理1～5可得以下结论：

引理7 对于如图2所示的图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 主要结果及其证明

证明 如图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，

证明 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，插秧机插植部两侧浮板的最大高度差可以达到6.1cm，插植部工作中期望最大倾角达到5.71°。在静态试验中秧苗插深自适应调节系统的控制相对误差均在5%以内，其中当该系统工作在半量程区间时，有着较好的控制精度。而系统处于初始以及接近满量程工作区间时，控制精度较差，相对误差在4%以上。究其原因，这应是该系统横向仿形控制机构的机械结构设计引起的。

+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.

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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

国家民委科研项目(14XBZ018)；甘肃省自然科学基金(145RJZA158)；西北民族大学科研创新团队计划资助项目；中央高校基本科研业务费专项资金项目(31920140059)．

柴文丽(1988— )，女，甘肃天水人，硕士研究生.

O157.5

A

1009-2102(2015)04-0001-05