给定片段数的树、单圈图和双圈图的极值p-谱半径.pdf
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1、第 43 卷第 3 期2023 年 9 月数学理论与应用MATHEMATICAL THEORY AND APPLICATIONSVol.43No.3Sep.2023The Extremalpspectral Radii of Trees,Unicyclic and Bicyclic Graphswith Given Number of SegmentsQiu MairongHe Xiaocong(School of Mathematics and Statistics,Central South University,Changsha 410083,China)AbstractLet G be
2、 a finite and simple graph.A walk S is called a segment of G if the endpoints(not necessarilydistinct)of S are of degree 1 or at least 3,and each of the rest vertices is of degree 2 in G.In this paper,we determinethe graphs that maximize the pspectral radius for p 1 among trees,unicyclic and bicycli
3、c graphs with given orderand number of segments,respectively.Key wordspspectral radiusTreeUnicyclic graphBicyclic graphSegment给定片段数的树、单圈图和双圈图的极值p谱半径邱买容贺晓聪*(中南大学数学与统计学院,长沙,410083)摘要设 G 是一个有限简单图.S 是 G 的一条途径.如果 S 的端点(可以相同)在 G 中的度是 1 或者至少是 3,且其他的顶点在 G 中的度都是 2,则称 S 为 G 的一个片段.本文对大于 1 的实数 p,分别确定固定阶数和片段数的树、
4、单圈图和双圈图的最大 p谱半径,并刻画对应的极图.关键词p谱半径树单圈图双圈图片段doi:10.3969/j.issn.1006 8074.2023.03.0031IntroductionAssume all graphs in this paper are finite and simple.Let G be a graph with vertex set V(G)=v1,v2,vn and edge set E(G).We follow the notation and terminologies in 1 except otherwisestated.If two vertices vi
5、and vjare adjacent,we write vi vj,otherwise,write vi vj.The adjacencymatrix of G is defined as A(G)=(aij)nnwhere aij=1 if vi vj,and aij=0 otherwise.Let dG(vi)be the degree of viand NG(vi)be the set of neighbors of viin G.As usual,denote by(G)the maximumdegree in G.The distance between two vertices u
6、 and v,denoted by d(u,v),is the length of a shortestpath between u and v.If H is a subgraph of G,let d(u,H)=mind(u,v)|v V(H).Denote by Pnand Cnthe path and the cycle on n vertices,respectively.k paths Pl1,Pl2,Plkaresaid to have almost equal lengths if|li lj|1 for 1 i,j k.A vertex of degree 1 in a gr
7、aph is saidThisworkissupportedbytheFundamentalResearchFundsfortheCentralUniversitiesofCentralSouthUniversity(No.2021zzts0034)Corresponding author:He Xiaocong Email:收稿日期:2022 年 10 月 31 日62数学理论与应用to be a pendant vertex and a vertex of degree at least 3 is called a branching vertex.An induced path with
8、vertices u1,u2,utis called a pendant path of G,if dG(u2)=dG(ut1)=2 and dG(ut)=1(note that there is no requirement on the degree of u1).A walk is a sequence of vertices u1,u2,ukofG such that ui1 uifor i=2,k,where the first vertex u1and last vertex ukare called endpoints.A walk S is a segment of a gra
9、ph if the endpoints(not necessarily distinct)of S are of degree 1 or at least3,and each of the rest vertices is of degree 2 in the graph.Let m=|E(G)|.If G is connected and m=n+c1,then G is called a ccyclic graph.Especially,if c=0,1 and 2,then G is called a tree,unicyclic graph and bicyclic graph,res
10、pectively.For c 1,Thebase of a ccyclic graph G,denoted byG,is the minimal ccyclic subgraph of G.It is easy to know thatG is the unique ccyclic subgraph of G containing no pendant vertices,while G can be obtained fromGby attaching trees to some vertices ofG.The pspectral radius was introduced for uni
11、form hypergraphs by Keevash,Lenz and Mubayi in 10andsubsequentlystudiedin5,8,17,18.ForasimplegraphG,letx=(xvi)Rn,wherexvicorrespondstothevertexvi.ThequadraticformofGisdefinedasPG(x):=2vivjE(G)xvixvj=xTA(G)x.Usingthe definition of quadratic form of graphs,for any real number p 1,the pspectral radius
12、of G is definedas(p)(G)=maxxp=1PG(x)=maxxp=12vivjE(G)xvixvj.A vector x Rnsatisfying xp=1 and(p)(G)=PG(x)is called an eigenvector to(p)(G).Forany p 1,if x is a vector such that xp=(ni=1|xvi|p)1p=1 and(p)(G)=PG(x),then thevector x=(|xvi|)also satisfies xp=1,and so(p)(G)=PG(x)PG(x)(p)(G)implies that(p)
13、(G)=PG(x).Therefore,there is always a nonegative vector x such that xp=1 and(p)(G)=PG(x).Let x be an eigenvector to(p)(G).If p 1,by Lagranges method,one can show that(p)(G)xp1vk=viNG(vk)xvifor k=1,2,n,which is called the eigenequation of(p)(G)for the vertex vk.The pspectral radius(p)(G)is a multifac
14、eted parameter,as(1)(G)/2 is the Lagrangian of G(see 19).Especially,(2)(G)is the usual spectral radius introduced by Cooper and Dutle 4,andlimp(p)(G)=2|E(G)|(see 17).The investigation on the spectral radius of graphs is an importanttopic in the theory of graph spectra.Recently,the problem concerning
15、 graphs with maximal or minimalspectral radius of a given class of graphs has been studied extensively.Brualdi and Solheid 2 studiedthe spectral radius of connected graphs.Guo 7 determined graphs with the largest spectral radius amongall the unicyclic graphs and all bicyclic graphs on n vertices and
16、 k pendant vertices,respectively.Lin andZhou 13 obtained the structure of trees that maximize the distance spectral radius among trees with givenorder and number of segments.Lin and Song 16 considered the family of all trees with given number ofsegment sequences and proved that the generalized star
17、minimizes the Wiener index.It should be notedthat extremal problems for(p)(G)mainly focus on hypergraphs(see 4,5,8,9,11,17,18).Kang,Liu,给定片段数的树、单圈图和双圈图的极值 p谱半径63Lu and Wang 8 characterized the 3uniform hypergraphs with maximum pspectral radius for p 1among the BergeG hypergraphs when G is a path,a c
18、ycle or a star.For more advances in research onspectral radius of graphs,we refer the readers to 3,6,7,12 15.Let k be the number of segments in G and be the number of vertices of degree 2 in G.If S is asegment of G,we write S G.Denote the length of a segment S by lSand the number of vertices ofdegre
19、e 2 in S by S.It is easy to know thatSGlS=m(G)and S=lS 1.Then=SGS=SG(lS 1)=SGlS k=m(G)k.Therefore,in the class of ccyclic nvertex graphs,as long as the number of segments of graphs is given,the number of vertices of degree 2 is determined.Based on this observation,in this paper,we determinethe struc
20、ture of trees,unicyclic and bicyclic graphs attaining the maximum pspectral radius with givennumber of segments,respectively.Let G(n,c,k),T(n,k),U(n,k)and B(n,k)be the set of ccyclic graphs,trees,unicyclic graphs andbicyclic graphs with n vertices and k segments,respectively.Obviously,T(n,1)=Pnand T
21、(n,0)=T(n,2)=U(n,0)=Cnand U(n,1)=B(n,0)=B(n,1)=.Thus,we only need to considertrees in T(n,k)for 3 k n1,unicyclic graphs in U(n,k)for 2 k n,bicyclic graphs in B(n,k)for 2 k n+1.Let Tn,kbe an nvertex tree with exactly one branching vertex formed by identifying one endvertexof each of the k paths with
22、almost equal lengths(see Figure 1 for it and all the following graphs).Let(i,j,k)be an nvertex unicyclic graph obtained from C3by attaching i,j and k pendant vertices to eachvertex of C3,respectively.We write knfor a unicyclic graph on n vertices obtained from C3by attachingk paths of almost equal l
23、engths to one vertex.Let Bbe a bicyclic graph with 4 vertices and Bbe abicyclic graph with 5 vertices and exactly one branching vertex.Let Bkbe a bicyclic graph obtained byattaching k pendant paths of almost equal lengths to one vertex of degree 3 in B.Let Bkbe a bicyclicgraph obtained by attaching
24、k pendant paths of almost equal lengths to the vertex of degree 4 in B.LetH1be attained from Bby attaching k7 pendant vertices to one vertex of degree 3 of B,and attachinga pendant vertex to the vertices of degree 2 in B,respectively.Let H2be attained from Bby attachingk 5 pendant vertices to one ve
25、rtex of degree 3 of B,and attaching a pendant vertex to one vertex ofdegree 2 in B.We can now state our main results.Theorem 1.1Let p 1,3 k n 1 and T T(n,k).Then(p)(T)1.For any U U(n,k),we have(1)if n=k and k 6,then(p)(U)(p)(k 5,1,1),unless U=(k 5,1,1)(2)if n=k+1 and k 4,then(p)(U)(p)(k 3,1,0),unles
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