{"title":"关于 k $k$ 树的子 k $k$ 树的最大局部平均阶数","authors":"Zhuo Li, Tianlong Ma, Fengming Dong, Xian'an Jin","doi":"10.1002/jgt.23128","DOIUrl":null,"url":null,"abstract":"<p>For a <span></span><math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-tree <span></span><math>\n <semantics>\n <mrow>\n <mi>T</mi>\n </mrow>\n <annotation> $T$</annotation>\n </semantics></math>, a generalization of a tree, the local mean order of sub-<span></span><math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-trees of <span></span><math>\n <semantics>\n <mrow>\n <mi>T</mi>\n </mrow>\n <annotation> $T$</annotation>\n </semantics></math> is the average order of sub-<span></span><math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-trees of <span></span><math>\n <semantics>\n <mrow>\n <mi>T</mi>\n </mrow>\n <annotation> $T$</annotation>\n </semantics></math> containing a given <span></span><math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-clique. The problem whether the maximum local mean order of a tree (i.e., a 1-tree) at a vertex is always taken on at a leaf was asked by Jamison in 1984 and was answered by Wagner and Wang in 2016. Actually, they proved that the maximum local mean order of a tree at a vertex occurs either at a leaf or at a vertex of degree 2. In 2018, Stephens and Oellermann asked a similar problem: for any <span></span><math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-tree <span></span><math>\n <semantics>\n <mrow>\n <mi>T</mi>\n </mrow>\n <annotation> $T$</annotation>\n </semantics></math>, does the maximum local mean order of sub-<span></span><math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-trees containing a given <span></span><math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-clique occur at a <span></span><math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-clique that is not a major <span></span><math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-clique of <span></span><math>\n <semantics>\n <mrow>\n <mi>T</mi>\n </mrow>\n <annotation> $T$</annotation>\n </semantics></math>? In this paper, we give it an affirmative answer.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 2","pages":"393-409"},"PeriodicalIF":0.9000,"publicationDate":"2024-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the maximum local mean order of sub-\\n \\n \\n k\\n \\n $k$\\n -trees of a \\n \\n \\n k\\n \\n $k$\\n -tree\",\"authors\":\"Zhuo Li, Tianlong Ma, Fengming Dong, Xian'an Jin\",\"doi\":\"10.1002/jgt.23128\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>For a <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math>-tree <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>T</mi>\\n </mrow>\\n <annotation> $T$</annotation>\\n </semantics></math>, a generalization of a tree, the local mean order of sub-<span></span><math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math>-trees of <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>T</mi>\\n </mrow>\\n <annotation> $T$</annotation>\\n </semantics></math> is the average order of sub-<span></span><math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math>-trees of <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>T</mi>\\n </mrow>\\n <annotation> $T$</annotation>\\n </semantics></math> containing a given <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math>-clique. The problem whether the maximum local mean order of a tree (i.e., a 1-tree) at a vertex is always taken on at a leaf was asked by Jamison in 1984 and was answered by Wagner and Wang in 2016. Actually, they proved that the maximum local mean order of a tree at a vertex occurs either at a leaf or at a vertex of degree 2. In 2018, Stephens and Oellermann asked a similar problem: for any <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math>-tree <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>T</mi>\\n </mrow>\\n <annotation> $T$</annotation>\\n </semantics></math>, does the maximum local mean order of sub-<span></span><math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math>-trees containing a given <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math>-clique occur at a <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math>-clique that is not a major <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math>-clique of <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>T</mi>\\n </mrow>\\n <annotation> $T$</annotation>\\n </semantics></math>? In this paper, we give it an affirmative answer.</p>\",\"PeriodicalId\":16014,\"journal\":{\"name\":\"Journal of Graph Theory\",\"volume\":\"107 2\",\"pages\":\"393-409\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-06-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Graph Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23128\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23128","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
On the maximum local mean order of sub-
k
$k$
-trees of a
k
$k$
-tree
For a -tree , a generalization of a tree, the local mean order of sub--trees of is the average order of sub--trees of containing a given -clique. The problem whether the maximum local mean order of a tree (i.e., a 1-tree) at a vertex is always taken on at a leaf was asked by Jamison in 1984 and was answered by Wagner and Wang in 2016. Actually, they proved that the maximum local mean order of a tree at a vertex occurs either at a leaf or at a vertex of degree 2. In 2018, Stephens and Oellermann asked a similar problem: for any -tree , does the maximum local mean order of sub--trees containing a given -clique occur at a -clique that is not a major -clique of ? In this paper, we give it an affirmative answer.
期刊介绍:
The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences.
A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .