给定直径的单圈图的超Wiener指数

The hyper-Wiener index is a kind of extension of the Wiener index, used for predicting physicochemical properties of organic compounds. The hyper-Wiener index W W(G) is defined as WW(G) =1/2∑_(u,v)∈V(G)(d_G(u, v) + d_G~2(u,v)) with the summation going over all pairs of vertices in G, d_G(u,v) denotes the distance of the two vertices u and v in the graph G. In this paper,we study the minimum hyper-Wiener indices among all the unicyclic graph with n vertices and diameter d, and characterize the corresponding extremal graphs.     

一类三圈图的Wiener指数

图G的Wiener指数是指图G中所有顶点对间的距离之和,即W（G）=∑dc（u,u）,{u,u}CG其中de（u,u）表示G中顶点u,u之间的距离.三圈图是指边数与顶点数之差等于2的连通图,任意两个圈至多只有一个公共点的三圈图记为T_n~3.研究了三圈图T_n~3的Wiener指数,给出了其具有最小、次小Wiener指数的图结构.     

仙人掌图的Wiener极化指数

图的Wiener极化指数定义为图中距离为3的无序点对的数目.本文给出仙人掌图的Wiener极化指数的显示表示,并导出若干特殊仙人掌图的极化指数公式.     

直径为4的树的IC-着色和IC-指数

根据Salehi等人在Discrete Mathematics上提出的图的IC-指数及极大IC-着色的相关概念,研究了直径为4的树T=T(m_1,m_2,…,m_s)的IC=着色问题·得到了当2≤<_1,m_2,…,m_s-1≤m_s,s≥2时,树T的IC-指数为Π_j=1~s(2~mj+1)+(2m,+1),其极大IC-着色有|π|种,其中|π|为m_1,同_2,…m_…s-1的全排列数.这为确定图的IC-指数提供了一般方法.     

具有固定控制数的树的调和指数

为了预测分子的物理、化学性质和生物活性, 科学家们提出了许多拓扑指数。调和指数是著名的Randić指数的一种变形形式, 研究表明该指数能有效地预测化合物的物理化学性质。对具有固定控制数的树的调和指数进行了研究, 通过分析具有固定控制数的树的结构, 利用数学归纳法, 给出了具有固定控制数的树的调和指数的最大值和最小值, 并刻画了达到最值的树。     

一类新的等可分树和化学树

设T是一个n阶树,e是它的一条边.用n1(e T)和n2(e T)分别表示树T中位于边e两侧的顶点的个数;n1(e T)+n2(e T)=n.设T和T′都是n阶树,e为T的一条边,f为T′的一条边,且n1(e T)=n1(f T′)或者n1(e T)=n2(f T′),则称e和f是等可分的边;如果能适当排列T的边e1,e2,…,en-1和T′的边e1′,e2′,…,en-′1,使得ei和ei′(i=1,2,…,n-1)都是等可分边,则称T和T′是等可分的树.等可分的化学树具有相同的W iener指数,因而有相似的物理化学性质.I.G u tm an等人给出了一些方法,构造等可分的树和化学树.本文给出了一种方法,构造出了一类新的等可分树和化学树.     

若干图类的Wiener指数的极值

一个图的Wiener指数是指这个图中所有点对的距离和.Wiener指数在理论化学中有广泛应用. 本文刻画了给定顶点数及特定参数如色数或团数的图中Wiener指数达最小值的图, 同时也刻画了给定顶点数及团数的图中Wiener指数达最大值的图.     

直径为d的n阶树的谱半径

二分图的特征值在量子化学中有意义，因此研究其图论性质和其特征值间的关系是有背景的，设Pd＋1（[d＋2/2],n-d-1,Pd＋1（[d＋4/2],n-d-1）分别为路Pd＋1的第[d＋2/2]和第[d＋4/2]个顶点上接出n-d-1条悬挂边所得到的树，本文证明了：若把所有直径为d（d≥1）的n阶树按其最大特征值从大到小的顺序排列，则排在前两位的依次是Pd＋1（[d＋2/2],n-d-1,Pd＋1（[d＋4/2],n-d-1） 。     

冠图(G1 (o) G2)(o) G3的Wiener指数

给出了冠图(G1 (o)G2)(o) G3的Wiener指数W((G1(o)G2)(o) G3),得出了W((G1 (o)C2)(o)G3)与三个图的边数和顶点数有关,且与G1的结构有关,而与G2、G3的结构无关.     

直径为4的整树

本文给出了直径为4的整树的一个充分条件,并由此给出了直径为4的整树的许多新类．最后提出一些基本的公开问题．     

Binary trees with the largest number of subtrees

This paper characterizes binary trees with n leaves, which have the greatest number of subtrees. These binary trees coincide with those which were shown by Fischermann et al. [Wiener index versus maximum degree in trees, Discrete Appl. Math. 122(1-3) (2002) 127-137] and Jelen and Triesch [Superdominance order and distance of trees with bounded maximum degree, Discrete Appl. Math. 125 (2-3) (2003) 225-233] to minimize the Wiener index.     

Equiseparability on terminal Wiener index

The aim of this work is to explore the properties of the terminal Wiener index, which was recently proposed by Gutman et al. (2004) [3], and to show the fact that there exist pairs of trees and chemical trees which cannot be distinguished by using it. We give some general methods for constructing equiseparable pairs and compare the methods with the case for the Wiener index. More specifically, we show that the terminal Wiener index is degenerate to some extent.     

Doubly stochastic graph matrices, II

If G is a graph on n vertices, its Laplacian matrix L(G) = D(G) - A(G) is the difference of the diagonal matrix of vertex degrees and the adjacency matrix. The main purpose of this note is to continue the study of the positive definite, doubly stochastic graph matrix (I_n + L(G))-¹= ω(G) = (w_ij). If, for example, w(G) = min w_ij, then w(G)≥0 with equality if and only if G is disconnected and w(G) ≤ l/(n + 1) with equality if and only if G = K_n. If i¦j, then w_ii ≥2wij, with equality if and only if the ith vertex has degree n - 1. In a sense made precise in the note, max w,, identifies most remote vertices of G. Relations between these new graph invariants and the algebraic connectivity emerge naturally from the fact that the second largest eigenvalue of ω(G) is 1/(1 + a(G)).     

积图的Steiner k-hyper Wiener指标

令图G是一个连通图。当 2leqslant kleqslant n-1时, 图G的Steiner k-hyper Wiener指标定义为 {rm SWW}_{k}(G)=frac{1}{2}sum_{Ssubseteq V (G), S=k}d_{G}(S)+frac{1}{2}sum_{Ssubseteq V (G), S=k}d_{G}(S)^{2}, 其中 d_{G}(S)表示图G中S的Steiner距离, 即连通图G中包含点集S的最小连通子图的边数。本文中我们确定了连图和字典积图的Steiner k-hyper Wiener指标的表达式, 给出了笛卡尔积图, 聚类图和冠状图的Steiner k-hyper Wiener指标的下限。     

Hosoya指标在一个闭区间的一类树的刻画(英文)

Let n and d be two positive integers.By B_n,d we denote the graph obtained by identifying an endvertex of path P_d with the center of star S_n-d+1,where n ≥ d + 1.By C_n,d we denote the graph obtained by identifying an endvertex of P_d-1 with the center of Stare S_n-d,and the other endvertex of P_d-1 with the center of S₃ where n ≥ d + 3.By E_n,d,k we denote the graph obtained by identifying the vertex v_k of P(v₁ - v₂ - ··· - v_d+1) with the center of S_n-d.In this paper,we completely characterize all trees T which have diameter at least d（d ≥ 3） and satisfy the following conditions:（i） Z(B_n,d) ≤ Z（T） ≤ Z(E_n,d,3) for n = d + 3;（ii） Z(B_n,d) ≤ Z（T） ≤ Z(C_n,d) for n ≥ d + 4.     

图的Hyper-Wiener指数的综述(英文)

设G是一个图.G的顶点u和v的距离是u和v之间最短路的长度.Wiener指数是G中所有无序顶点对之间距离之和,而Hyper-Wiener指数定义为WW(G)=?∑u,v∈V(G)d(u,v)+?∑u,v∈V(G)d2(u,v),式中的和取遍G的所有顶点对.本文总结了图的Hyper-Wiener指数的最近结论.