具有两个度数的树（英文）_乔璞.pdf

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 is 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 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 code: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)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 connected 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 called 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: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 examples:(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(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 theoreticproperties.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 -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 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 degree 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 andonly 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 .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 the path .Next assume .3d n+1(n 2)/(d 1)3(n 2)/(d 1)=1d=n 1Sn(1,n 1)n(n 2)/(d 1)=2d=n/2CP(n,n/2)(1,n/2)n(n 2)/(d 1)=3x,y,zd(n+1)/3x,yzT3d=n+1xyzxyTT=CP(n,d)Suppose .Then .There are three possibilities.If ,i.e.,then the star is the unique -tree of order .If ,i.e.,then the caterpillar (a double-broom)is a unique -tree of order .If,let be the three vertices of degree which equals .If any two ofthe vertices ,and do not have a common neighbor,then the tree would have order at least.This constitutes a contradiction.Without loss of generality,suppose and have acommon neighbor,which must be .Now,and are nonadjacent since contains no cycles.Itfollows that and the uniqueness is proved.3d n(1,d)nCP(n,d)d 1n 2d 1(n d+1)2CP(n d+1,d)Gv1,v2,vkdGk=(n d+1)2)/(d 1)k3d nk 3H(n,d)Gd 1v2CP(17,4)H(17,4)Now,suppose .We will show that there exist at least two non-isomorphic -trees oforder .First,the caterpillar is such a tree.The condition that divides implies divides .Hence,by Theorem 1,the caterpillar ,which we denoteby ,exists.Let be the vertices of degree on the spine of in order where.Since is an integer,the assumption implies that .Let be the graph obtained from by attaching edges to one of the leaf neighbors of .The graphs and are depicted in Figure 2.Fig.2 CP(17,4)and H(17,4)CP(n,d)H(n,d)(1,d)n(n 2)/(d 1)+1(n 2)/(d 1)Both and are -trees of order ,but they are non-isomorphic,since theformer has diameter while the latter has diameter .The proof isthus complete.Finally,we consider the maximum possible diameter of non-path two-degree trees of a given order.n 4pn 2n(n 2)/p+1Theorem 3 Let be a positive integer and suppose is the smallest prime divisor of .Then,the maximum possible diameter of a non-path two-degree tree of order is .(1,d)d=2T(1,d)nd 3TkP:x0,x1,xkTx0 xkkxiP1 i k 1dn(k+1)+(k 1)(d 2)k (n 2)/(d 1)+1CP(n,d)(n 2)/(d 1)+1(1,d)n(n 2)/(d 1)+1nProof First,note that a -tree of order at least three is a path if and only if .Let bea -tree of order with .Suppose has diameter and let be a diame-tral path of ;hence,the distance between and is .Since each internal vertex of with has degree and any two of them do not have a common neighbor,we have .Hence,.On the other hand,the caterpillar has diameter .This proves that the maximum diameter of a -tree of order is .It follows that the maximum possible diameter of a non-path two-degree tree oforder is第 2 期乔 璞,等:具有两个度数的树(英)3maxn 2d 1+1|d 3,d 1 divides n 2=n 2p+1,pn 2where is the smallest prime divisor of .The following corollary is an immediate consequence of Theorem 3.n 4nn/2Corollary 2 If is an even positive integer,then the maximum possible diameter of a non-path two-degree tree of order is .References BONDY J A,MURTY U S R.Graph Theory M.New York:Springer,2008.1 BLASS A,HARARY F,MILLER Z.Which trees are link graphs?J.Journal of Combinatorial Theory(Series B),1980,29:277-292.2 BROERSMA H,XIONG L,YOSHIMOTO K.Toughness and hamiltonicity in k-trees J.Discrete Math,2007,307:832-838.3(责任编辑:陈丽贞)4华东师范大学学报(自然科学版)2023 年