ArticleID：1000-5641(2023)02-0001-04Two-degreetreesQIAOPu1,ZHANXingzhi2(1.SchoolofMathematics,EastChinaUniversityofScienceandTechnology,Shanghai200237,China;2.SchoolofMathematicalSciences,EastChinaNormalUniversity,Shanghai200241,China)1dd⩾2;(1,d)n,d(1,d)n;d(1,d)nn.Abstract:Agraphiscalledatwo-degreegraphifitsverticeshaveonlytwodistinctdegrees.Atwo-degreetreeoforderatleastthreehavetwodegrees,andforsomesuchatreeiscalleda-tree.Givenapositiveintegerwedetermine:(1)thepossiblevaluesofsuchthatthereexistsa-treeoforder(2)thevaluesofsuchthatthereexistsaunique-treeoforder,and(3)themaximumdiameteroftwo-degreetreesoforderTheresultsprovideanewexampleshowingthatthebehaviorofgraphsmaysometimesbedeterminedbynumbertheoreticproperties.Keywords:two-degreetree;diameter;uniquegraphCLCnumber:O157.5Documentcode:ADOI:10.3969/j.issn.1000-5641.2023.02.001具有两个度数的树乔璞1,詹兴致2（1.华东理工大学数学学院,上海200237;2.华东师范大学数学科学学院,上海200241）dd(1,d)nn(1,d)dn(1,d)dn(1,d)摘要:如果一个图只有两个不同的度数,这个图就称为二度图.阶数至少为3的二度树具有度数1和,这里是至少为2的整数,这样的树称为-树.给定一个正整数,确定了以下信息:(1)存在一个阶-树的可能的的值;(2)存在唯一的阶-树的可能的的值;(3)阶-树的最大可能直径.这些结果提供了一个新的例子,表明有时候图的行为是由数论性质决定的.关键词:二度树;直径;唯一图1dd⩾2;(1,d)(1,3)1(1,d)ddTheorderofagraphisitsnumberofvertices,andthesizeofagraphisitsnumberofedges.Atreeisaconnectedgraphthatcontainsnocycles.Treesarethesimplestconnectedgraphsinthesensethattheyhavetheleastpossiblesizeamongconnectedgraphsofagivenorder.Agraphiscalledatwo-degreegraphifitsverticeshaveonlytwodistinctdegrees.Aleafofagraphisavertexofdegree1.Everynontrivialtreehasatleasttwoleaves[1].Thus,atwo-degreetreehastwodegrees,andforsomesuchatreeiscalleda-tree.-treesarecalledcubictrees[2].Onemightthinkthatwecanomit“”andsimplyrefertoa-treeasa-tree.However,thiscannotbedone,becausetheterm-treealreadyhasanotherwellestablishedmeaning[3].nd(1,d)Foragivenpositiveinteger,whatarethepossiblevaluesofsuc...
