给定分数匹配数的图的(距离)无符号Laplace谱半径  

(Distance)signless Laplacian spectral radius of graphs with fixed fractional matching number

作　　者：常彩冰 刘岩[1] Caibing Chang;Yan Liu

机构地区：[1]华南师范大学数学科学学院,广州510631

出　　处：《中国科学：数学》2024年第11期1773-1786,共14页Scientia Sinica：Mathematica

基　　金：广州市科技计划(批准号:202002030183);广东省自然科学基金(批准号:2021A1515012045);国家自然科学基金(批准号:12161073)资助项目。

摘　　要：图G的分数匹配是指一个函数f:E(G)→[0,1],使得对于任意点v∈V(G),都有Σ_(e∈ΓG(v))f(e)6≤1,其中ΓG(v)表示G中与点v关联的边的集合.图G的分数匹配数μ_(f)(G)是指对所有分数匹配f,Σ_(e∈E(G))f(e)的最大值.刘岩和刘桂真(2002)给出了图G的分数匹配数μf(G)与匹配数(G)的关系式:μ_(f)(G)=μ(G)+(nc(G))/2.本文根据这个公式,刻画了在分数匹配数上饱和的图,其中饱和图是指一个图G,使得对于任意两个不相邻的点u和v,都有μ_(f)(G+uv)>μ_(f)(G).从而,在给定分数匹配数和点数的图集中,本文刻画具有最小距离无符号Laplace谱半径的极图,分别得到无符号Laplace谱半径的一个上界和一个下界,并刻画具有最大无符号Laplace谱半径的极图.A fractional matching of a graph G is a functionf:E(G)→[0,1]such that for each vertex v,Σ_(e∈ΓG(v))f(e)6≤1,whereΓG(v)is the set of edges incident with v.The fractional matching number μ_(f)(G) of G is the maximum value of ,Σ_(e∈E(G))f(e)over all fractional matchings f.Liu and Liu(2002)obtained the relationship between the fractional matching number and the matching number of a graph G which is thatμ_(f)(G)=μ(G)+(nc(G))/2.In this paper,we characterize the saturated graph with a given fractional matching number by using this formula,where the saturated graph is a graph G such that μ_(f)(G+uv)>μ_(f)(G) for any two nonadjacent vertices u and v of G.Among n-vertex graphs with given fractional matching number,we characterize the extremal graph that has the minimum distance signless Laplacian spectral radius,give an upper bound and a lower bound of signless Laplacian spectral radius,and characterize the extremal graph that has the maximum signless Laplacian spectral radius.

关 键 词：分数匹配数 饱和图 无符号 LAPLACE 谱半径 距离无符号 LAPLACE 谱半径 

分 类 号：O157.5[理学—数学]