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给定悬挂点个数的分子树的ISDD指数的极值
Extremal ISDD Index of Molecular Trees with Fixed Number of Pendent Vertices
- 中北大学学报(自然科学版) 2024年45卷第1期 页码：30-35
- 作者机构：
中北大学 数学学院，山西 太原 030051
- 作者简介：
赵芳方（1997-），女，硕士，主要从事图论、组合数学方向的研究。
邵燕灵（1963-），女，教授，博士，主要从事图论、组合数学方向的研究。E-mail：ylshao@nuc.edu.cn。
- 基金信息：山西省自然科学基金资助项目(201901D211227)
DOI：10.3969/j.issn.1673-3193.2024.01.004
中图分类号： O178收稿：2022-11-18，
纸质出版：2024-02-29
- 稿件说明：
移动端阅览
赵芳方， 邵燕灵. 给定悬挂点个数的分子树的ISDD指数的极值［J］. 中北大学学报（自然科学版）， 2024， 45（1）： 30-35.
ZHAO Fangfang， SHAO Yanling. Extremal ISDD index of molecular trees with fixed number of pendent vertices［J］. Journal of North University of China（Natural Science Edition）， 2024， 45（1）： 30-35.
赵芳方， 邵燕灵. 给定悬挂点个数的分子树的ISDD指数的极值［J］. 中北大学学报（自然科学版）， 2024， 45（1）： 30-35. DOI： 10.3969/j.issn.1673-3193.2024.01.004.
ZHAO Fangfang， SHAO Yanling. Extremal ISDD index of molecular trees with fixed number of pendent vertices［J］. Journal of North University of China（Natural Science Edition）， 2024， 45（1）： 30-35. DOI： 10.3969/j.issn.1673-3193.2024.01.004.
摘要
设
<math id="M1"><mi>G</mi><mo>=</mo><mfenced separators="|"><mrow><mi>V</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo>
</mo><mi>E</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812114&type=
4.06400013
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812115&type=
28.53266525
为
<math id="M2"><mi>n</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812132&type=
2.37066650
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812107&type=
1.60866666
阶连通图，其顶点集为
<math id="M3"><mi>V</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812108&type=
3.47133350
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812149&type=
8.46666718
，边集为
<math id="M4"><mi>E</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812150&type=
3.47133350
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812112&type=
8.04333401
，用
<math id="M5"><mi>d</mi><mi>e</mi><mi>g</mi><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812158&type=
3.47133350
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812140&type=
10.07533360
表示顶点
<math id="M6"><mi>x</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812161&type=
2.37066650
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812156&type=
1.94733346
的度，则图
<math id="M7"><mi>G</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812174&type=
2.37066650
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812203&type=
2.45533323
的反对称分割指数为
<math id="M8"><mi>I</mi><mi>S</mi><mi>D</mi><mi>D</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>x</mi><mi>y</mi><mo>∈</mo><mi>E</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced></mrow><mrow/></munderover></mstyle><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi>d</mi><mi>e</mi><mi>g</mi><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>⋅</mo><mi>d</mi><mi>e</mi><mi>g</mi><mfenced separators="|"><mrow><mi>y</mi></mrow></mfenced></mrow><mrow><mi>d</mi><mi>e</mi><mi>g</mi><msup><mrow><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>d</mi><mi>e</mi><mi>g</mi><msup><mrow><mfenced separators="|"><mrow><mi>y</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812205&type=
11.85333347
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812177&type=
56.30332947
。本文主要采用不等式和分类讨论法对具有固定悬挂点的分子树的
<math id="M9"><mi>I</mi><mi>S</mi><mi>D</mi><mi>D</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812193&type=
2.37066650
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812169&type=
7.78933382
指数进行了研究，分别讨论了悬挂点个数为偶数和悬挂点个数大于等于3时分子树的
<math id="M10"><mi>I</mi><mi>S</mi><mi>D</mi><mi>D</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812193&type=
2.37066650
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812169&type=
7.78933382
指数的极值，分子树是指顶点度不超过4的树。首先，确定了当悬挂点个数为偶数时，分子树中反对称分割指数为最小值，此时，
<math id="M11"><mi>I</mi><mi>S</mi><mi>D</mi><mi>D</mi><mfenced separators="|"><mrow><mi>M</mi><mi>T</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>n</mi><mo>-</mo><mfrac><mrow><mn mathvariant="normal">31</mn></mrow><mrow><mn mathvariant="normal">85</mn></mrow></mfrac><mi>p</mi><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">10</mn></mrow></mfrac></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812210&type=
6.77333355
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812172&type=
43.60333252
；其次，确定了当悬挂点个数大于等于3时，分子树中反对称分割指数为最大值，此时，
<math id="M12"><mi>I</mi><mi>S</mi><mi>D</mi><mi>D</mi><mfenced separators="|"><mrow><mi>M</mi><mi>T</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>n</mi><mo>-</mo><mfrac><mrow><mn mathvariant="normal">9</mn></mrow><mrow><mn mathvariant="normal">65</mn></mrow></mfrac><mi>p</mi><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812218&type=
6.77333355
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812213&type=
42.07933044
，并刻画了达到
<math id="M13"><mi>I</mi><mi>S</mi><mi>D</mi><mi>D</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812193&type=
2.37066650
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812169&type=
7.78933382
指数极值的分子树。
Abstract
Let
<math id="M14"><mi>G</mi><mo>=</mo><mfenced separators="|"><mrow><mi>V</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo>
</mo><mi>E</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812185&type=
4.74133301
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812187&type=
31.83466530
be a connected graph of order
<math id="M15"><mi>n</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812202&type=
2.79399991
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812263&type=
1.86266661
，
<math id="M16"><mi>V</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812264&type=
3.97933316
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812251&type=
9.82133293
be vertex set of
<math id="M17"><mi>G</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812228&type=
2.79399991
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812268&type=
2.87866688
，
<math id="M18"><mi>E</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812270&type=
3.97933316
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812271&type=
9.31333351
be edge set of
<math id="M19"><mi>G</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812228&type=
2.79399991
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812268&type=
2.87866688
，
<math id="M20"><mi>d</mi><mi>e</mi><mi>g</mi><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812272&type=
3.97933316
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812244&type=
11.00666618
be degree of the vertex
<math id="M21"><mi>x</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812259&type=
2.79399991
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812276&type=
2.20133328
. The inverse symmetric division deg index of
<math id="M22"><mi>G</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812228&type=
2.79399991
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812268&type=
2.87866688
is
<math id="M23"><mi>I</mi><mi>S</mi><mi>D</mi><mi>D</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>x</mi><mi>y</mi><mo>∈</mo><mi>E</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced></mrow><mrow/></munderover></mstyle><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi>d</mi><mi>e</mi><mi>g</mi><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>⋅</mo><mi>d</mi><mi>e</mi><mi>g</mi><mfenced separators="|"><mrow><mi>y</mi></mrow></mfenced></mrow><mrow><mi>d</mi><mi>e</mi><mi>g</mi><msup><mrow><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>d</mi><mi>e</mi><mi>g</mi><msup><mrow><mfenced separators="|"><mrow><mi>y</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812282&type=
13.71599960
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812283&type=
63.58466721
. Inequality and classification discussion are used to study the
<math id="M24"><mi>I</mi><mi>S</mi><mi>D</mi><mi>D</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812324&type=
3.55599999
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812286&type=
15.49400043
of molecular tree with fixed number of pendent vertices，respectively，the extreme value of the
<math id="M25"><mi>I</mi><mi>S</mi><mi>D</mi><mi>D</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812312&type=
2.79399991
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812313&type=
9.05933285
index of the molecular tree with the number of pendent vertices is even number and the number of pendent vertices is more than or equal to 3 are discussed， the tree whose vertex degree is less than 4 is called molecular tree. Firstly， the minimum value of the inverse symmetric division deg index of
<math id="M26"><mi>G</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812228&type=
2.79399991
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812268&type=
2.87866688
is determined when the number of pendent vertices is even， that is
<math id="M27"><mi>I</mi><mi>S</mi><mi>D</mi><mi>D</mi><mfenced separators="|"><mrow><mi>M</mi><mi>T</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>n</mi><mo>-</mo><mfrac><mrow><mn mathvariant="normal">31</mn></mrow><mrow><mn mathvariant="normal">85</mn></mrow></mfrac><mi>p</mi><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">10</mn></mrow></mfrac></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812290&type=
7.87400007
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812329&type=
50.88466644
. Secondly， when the number of pendent vertices is greater than or equal to 3， the maximum value of the inverse symmetric division deg index of
<math id="M28"><mi>G</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812228&type=
2.79399991
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812268&type=
2.87866688
in the molecular tree is determined， that is
<math id="M29"><mi>I</mi><mi>S</mi><mi>D</mi><mi>D</mi><mfenced separators="|"><mrow><mi>M</mi><mi>T</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>n</mi><mo>-</mo><mfrac><mrow><mn mathvariant="normal">9</mn></mrow><mrow><mn mathvariant="normal">65</mn></mrow></mfrac><mi>p</mi><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812330&type=
7.87400007
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812303&type=
49.10666656
， and the molecular tree of
<math id="M30"><mi>I</mi><mi>S</mi><mi>D</mi><mi>D</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812333&type=
2.79399991
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=96812322&type=
9.05933285
index reaching the extreme value is described.
关键词
Keywords
references
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