1、Article ID：1000-5641(2023)02-0001-04Two-degree treesQIAO Pu1,ZHAN Xingzhi2(1.School of Mathematics,East China University of Science and Technology,Shanghai200237,China;2.School of Mathematical Sciences,East China Normal University,Shanghai200241,China)1dd 2;(1,d)n,d(1,d)n;d(1,d)nn.Abstract:A graph i
2、s called a two-degree graph if its vertices have only two distinct degrees.A two-degreetree of order at least three have two degrees,and for some such a tree is called a -tree.Given a positive integer we determine:(1)the possible values of such that there exists a -tree oforder (2)the values of such
3、 that there exists a unique -tree of order ,and(3)the maximumdiameter of two-degree trees of order The results provide a new example showing that the behavior ofgraphs may sometimes be determined by number theoretic properties.Keywords:two-degree tree;diameter;unique graphCLC number:O157.5Document c
4、ode: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
5、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)ddThe order of a graph is its number of vertices,and the size of a graph is its number of edges.Atree is a connected graph that contains no cycles.Trees are the simplest connected graphs in the sensethat they have the least possible size among co
6、nnected graphs of a given order.A graph is called atwo-degree graph if its vertices have only two distinct degrees.A leaf of a graph is a vertex of degree1.Every nontrivial tree has at least two leaves1.Thus,a two-degree tree has two degrees,and for some such a tree is called a -tree.-trees are call
7、ed cubic trees2.One might thinkthat we can omit“”and simply refer to a -tree as a -tree.However,this cannot be done,because the term -tree already has another well established meaning3.nd(1,d)For a given positive integer ,what are the possible values of such that there exists a -收稿日期:2021-05-07基金项目:
8、国家自然科学基金(11671148,11771148);上海市科学技术委员会基金(18dz2271000)第一作者:乔璞,女,讲师,博士,研究方向为图论.E-mail:通信作者:詹兴致,男,教授,研究方向为图论、矩阵论.E-mail: 第 2 期华东师范大学学报(自然科学版)No.22023 年 3 月Journal of East China Normal University(Natural Science)Mar.2023n(1,d)20d=2,3,4,7,10,19,(1,d)39d=238d20,d39.tree of order?Let us consider two exampl
9、es:(1)There exists a -tree of order if and only if or and(2)there exists a -tree of order if and only if or .Thus,there are six possible values of for the order but there are only two possible values of for thelarger order Why?nd(1,d)nd(1,d)nnIn this paper,for a given positive integer ,we determine(
10、1)the possible values of such thatthere exists a -tree of order ;(2)the values of such that there exists a unique -tree oforder ;and(3)the maximum diameter of two-degree trees of order .These results provide a newexample showing that the behavior of graphs may sometimes be determined by number theor
11、eticproperties.A caterpillar is a tree in which a single path(the spine)is incident to or contains every edge;inother words,removal of its leaves yields a path.dnd 1n 2CP(n,d)n(1,d)Notation 1 Let and be positive integers such that divides .We use todenote the unique caterpillar of order which is a -
12、tree.CP(18,5)The caterpillar is depicted in Figure 1.Fig.1 The caterpillar CP(18,5)n 3(1,d)nd 1n 2Theorem 1 Let be a positive integer.Then,there exists a -tree of order if and onlyif divides .(1,d)nVeProof Suppose there exists a -tree of order .The degree sum formula1 proved byLeonhard Euler in 1736
13、 states that if a graph has vertex set and size ,thenxVdeg(x)=2e,deg(x)x(1,d)nkdn k1nn 1(n k)+kd=2(n 1)(d 1)k=n 2d 1n 2where denotes the degree of the vertex .Now suppose in a -tree of order ,there are vertices of degree and,hence,there are vertices of degree .Since a tree of order has size,by the d
14、egree sum formula we have ;alternatively,.Hence,divides .d 1n 2CP(n,d)(1,d)nConversely,suppose divides .Then,the caterpillar is a -tree of order .PnSnnWe denote by and the path and star of order ,respectively.PnSnn 4n 2Corollary 1 The path and the star are the only two-degree trees of order if andon
15、ly if is a prime number.n 21n 2n 2(1,d)nd 1=1d 1=n 2d=2d=n 1Pn(1,2)nSn(1,n 1)nProof If is a prime number,then and are the only divisors of .By Theorem 1,there exists a -tree of order if and only if or ;in other words,or.The path is the unique -tree of order and the star is the unique -tree of order
16、.n 2qn 22 q n 3(1,q+1)nPnSn3 q+1 n 2If is a composite number,let be a divisor of with .By Theorem 1,there exists a -tree of order ,which is neither the path nor the star since .2华东师范大学学报(自然科学版)2023 年(1,d)n 3d 1n 2d=23d n+1Theorem 2 There exists a unique -tree of order if and only if divides andfurthermore or .(1,d)Tnd 1n 2(n 2)/(d 1)d(1,2)Pnd 3Proof Suppose there exists a -tree of order .By Theorem 1 and its proof,divides,and is the number of vertices of degree .First,note that the only -treeis
