{"title":"On the maximum local mean order of sub-\u0000 \u0000 \u0000 k\u0000 \u0000 $k$\u0000 -trees of a \u0000 \u0000 \u0000 k\u0000 \u0000 $k$\u0000 -tree","authors":"Zhuo Li, Tianlong Ma, Fengming Dong, Xian'an Jin","doi":"10.1002/jgt.23128","DOIUrl":"10.1002/jgt.23128","url":null,"abstract":"<p>For a <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-tree <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math>, a generalization of a tree, the local mean order of sub-<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-trees of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math> is the average order of sub-<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-trees of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math> containing a given <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </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>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-tree <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math>, does the maximum local mean order of sub-<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-trees containing a given <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-clique occur at a <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-clique that is not a major <span></span","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 2","pages":"393-409"},"PeriodicalIF":0.9,"publicationDate":"2024-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141273372","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}