边无关数为q的n阶树的谱半径的第三大值  

The Third Largest Value of Spectral Radius of Trees with n Vertices and Edge-independence Number q

作　　者：郭曙光[1] 

机构地区：[1]南京师范大学数学与计算机科学学院

出　　处：《南京大学学报（自然科学版）》2004年第1期75-82,共8页Journal of Nanjing University（Natural Science）

基　　金：国家自然科学基金项目(10171046)

摘　　要：　设G为有限无向简单图,G的邻接矩阵的特征值称为G的特征值,G的最大特征值称为G的谱半径.二分图的特征值在量子化学中有意义,因而研究二分图的特征值有重要的实用价值.Kl1,k(k≥l≥1)记星图K1,k的l个悬挂点各接出一条悬挂边所得的图.Tn(q)表示边无关数为q(≥5)的n阶树的集合.(1,1)T(q-3,n-2q+1)∈Tn(q)为Kq-21,n-q-1的某个2度顶点上接出一条路P2所得的图.给出了Tn(q)中树的谱半径的第三大值,并证明了:当n-2q=1时,取得该值的唯一的树为Kq1,q;当n-2q≥2时,取得该值的树为(1,1)T(q-3,n-2q+1).All graphs are finite undirected graphs without loops and multiple edges. The eigenvalues of a graph G are the eigenvalues of its adjacency matrix. The spectral radius ρ(G) of a graph G is the largest eigenvalue of G. Eigenvalues of bipartite graphs have significance in quantum chemistry. Therefore it is of great practical value to study the eigenvalues of bipartite graphs. Denote by K^l_(1,k)(k≥l≥1) the graph obtained from the star K_(1,k) by joining a new vertex to each of l pendant vertices of K_(1,k). Let T_n(q) be the set of trees with n vertices and edge independence number q(q≥5). Denote by (()_((0,2))T_((q-2,n-2q)))∈T_n(q) the graph obtained from the K^(q-1)_(1,n-q) by joining a new vertex to one of the vertices of degree 2 of K^(q-1)_(1,n-q). In recent papers, previous workers gave the first and second largest values of spectral radius of trees in T_n(q) and proved that the only tree which reached that value was K^(q-1)_(1,n-q) or ()_((0,2))T_((q-2,n-2q)). Denote by (()_((1,1))T_((q-3,n-2q+1))∈)T_n(q) the graph obtained from the K^(q-2)_(1,n-q-1) by joining a path P_2 to one of the vertices of degree 2 of K^(q-2)_(1,n-q-1). Let q≥5,T∈T_n(q),TK^(q-1)_(1,n-q), ()_((0,2))T_((q-2,n-2q)) . This paper proves that if (n-2q=1,) then (ρ(T)≤q+1) and the equality holds if and only if TK^q_(1,q), and that if n-2q≥2, then (ρ(T)≤)(ρ(()_((1,1))T_((q-3,n-2q+1)))) and the equality holds if and only if T()_((1,1))T_((q-3,n-2q+1)), where (ρ(()_((1,1))T_((q-3,n-2q+1)))) is the largest root of the equation x^8-(n-q+3)x^6+(4n-5q+1)x^4-(4n-7q+3)x^2+n-2q+1=0.

关 键 词：边无关数 树 谱半径 邻接矩阵 图 

分 类 号：O151.21[理学—数学] O157.5[理学—基础数学]