三圈图的Harary指数

图G的Harary指数是指图G中所有顶点对间的距离倒数之和. 三圈图是指边数等于顶点数加2的连通图. 研究了三圈图的Harary数, 给出了所有三圈图中具有极大Harary指数的图的结构以及含有三个圈的三圈图中具有次大Harary指数的图的结构.     

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.     

强连通竞赛图的圈分解

设T为n阶强连通竞赛图．本文通过详细刻画不能进行圈分解的强连通竞赛图的特征，证明了满足max{^ ，δ^-}≥5k-5和k≥2的强连通竞赛图T,能够分解为k个圈．     

关于三圈连通标号图的计数公式

关于三圈连通标号图的计数公式张树生江西宁都固厚中学本文所指的图者是无向简单图。如果一个图恰好包含有ｍ个初级圈，那么就说这个图恰好包含有ｍ个单个的圈。Ｈａｒａｒｙ在［１］中提出了给定圈的个数的连通标号圈的计数问题。Ｒｅｎｙｉ在［２］中解决了单圈边通标号...     

上连通和超连通三次点传递图

图G称为上连通的,若对每个最小割集C,G-C有孤立点,G称为超连通的,若对每个最小割集C,G-C恰有两个连通分支,且其中之一为弧立点,本文刻划了上连通和超连通三次点传递图。     

3—连通正则无爪图的Hamilton圈

本文证明了：任一阶数不超过６ｋ－４的３－连通ｋ－正则无爪图是Ｈａｍｉｔｏｎ的。     

随机图的广义3-连通度

图的广义连通度的概念是由Chartrand等人引入的.令S表示图G的一个非空顶点集,κ(S)表示图G中连结S的内部不交树的最大数目.那么,对任意一个满足2≤r≤n的整数r,定义G的广义r-连通度为所有κ(S)中的最小值,其中S取遍G的顶点集合的r-元子集.显然,κ_2(G)=κ(G),即为图G的顶点连通度.所以广义连通度是经典连通度的一个自然推广.本文研究了随机图的广义3-连通度,证明了对任一给定的整数k,k≥1,p=(log n+(k+1)log long n-log lon logn)/n是关于性质κ_3(G(n,p))≥k的紧阈值函数.我们得到的结果可以看作是Bollobas和Thomason给出的关于经典连通度结果的推广.     

上连通和超连通三次点传递图(英)

图G称为上连通的,若对每个最小割集C,G-C有孤立点.G称为超连通的,若对每个最小割集G,G-C恰有两个连通分支,且其中之一为孤立点.本文刻划了上连通和超连通三次点传递图.     

2-连通半无爪图的最长圈

A graph G is called quasi-claw-free if it satisfies the property:d(x,y)=2 there exists a vertex u∈N(x)∩N(y)such that N[u]■N[x]∪N[y].In this paper,we show that every 2-connected quasi-claw-free graph of order n with G■F contains a cycle of length at least min{3δ+2,n},where F is a family of graphs.     

关于广义三项指数和的四次均值

本文利用三角和等式以及既约剩余系的性质, 研究了广义三项指数和的四次均值的计算问题, 并给出了特定情形下的一些等式.     

On the Laplacian integral tricyclic graphs

A graph is called Laplacian integral if all its Laplacian eigenvalues are integers. In this paper, we give an edge subdividing theorem for Laplacian eigenvalues of a graph (Theorem 2.1) and characterize a class of k-cyclic graphs whose algebraic connectivity is less than one. Using these results, we determine all the Laplacian integral tricyclic graphs. Furthermore, we show that all the Laplacian integral tricyclic graphs are determined by their Laplacian spectra.     

三圈图的无符号拉普拉斯谱半径

假设图G的点集是V(G)={v_1,v_2,…,v_n},用d_(v_i)(G)表示图G中点v_i的度,令A(G)表示G的邻接矩阵,D(G)是对角线上元素等于d_(v_i)(G)的n×n对角矩阵,Q(G)=D(G)+A(G)是G的无符号拉普拉斯矩阵,Q(G)的最大特征值是G的无符号拉普拉斯谱半径.现确定了所有点数为n的三圈图中无符号拉普拉斯谱半径最大的图的结构.     

On the spectral radius of tricyclic graphs with a fixed diameter

A tricyclic graph is a connected graph with n vertices and n + 2 edges. In this article, we determine graphs with the largest spectral radius among the n-vertex tricyclic graphs with given diameter d.     

The Hamiltonian index of graphs

The Hamiltonian index of a graph G is defined as

h(G)=min{m:L^m(G) is Hamiltonian}.     

The optimal general upper bound for the distinguishing index of infinite graphs

The distinguishing index $D\prime \left(G\right)$ of a graph $G$ is the least cardinal number $d$ such that $G$ has an edge-coloring with $d$ colors, which is preserved only by the trivial automorphism. We prove a general upper bound $D\prime \text{◂≤▸}\left(G\right)\le \mathrm{\Delta }-1$ for any connected infinite graph $G$ with finite maximum degree $\mathrm{\Delta }\ge 3$. This is in contrast with finite graphs since for every $\mathrm{\Delta }\ge 3$ there exist infinitely many connected, finite graphs $G$ with $\text{◂,▸}D\prime \left(G\right)=\mathrm{\Delta }$. We also give examples showing that this bound is sharp for any maximum degree $\mathrm{\Delta }$.