X. Deng et al. proved Chvātal＇s conjecture on maximal stable sets and maximal cliques in graphs. G. Ding made a conjecture to generalize Chvátal＇s conjecture. The purpose of this paper is to prove this conjecture in planar graphs and the complement of planar graphs.
A graph has exactly two main eigenvalues if and only if it is a 2-walk linear graph.In this paper,we show some necessary conditions that a 2-walk（a,b）-linear graph must obey.Using these conditions and some basic theorems in graph theory,we characterize all 2-walk linear graphs with small cyclic graphs without pendants.The results are given in sort on unicyclic,bicyclic,tricyclic graphs.
Generalized hypercubes （denoted by Q（d1,d2,... ,dn）） is an important network topology for parallel processing computer systems. Some methods of forming big cycle from small cycles and links have been developed. Basing on which, we has proved that in generalized hypercubes, every edge can be contained on a cycle of every length from 3 to IV（G）I inclusive and all kinds of length cycles have been constructed. The edgepanciclieity and node-pancilicity of generalized hypercubes can be applied in the topology design of computer networks to improve the network performance.
As an enhancement on the hypercube Qn, the augmented cube AQn, pro- posed by Choudum and Sunitha [Choudum S.A., Sunitha V., Augmented cubes, Networks, 40（2）（2002）, 71-84], possesses some properties superior to the hypercube Qn. In this paper, assuming that （u, v） is an arbitrary fault-free d-link in an n-dimensional augmented cubes, 1 ≤ d ≤ n - 1, n ≥ 4. We show that there exists a fault-free Hamiltonian cycle in the augmented cube contained （u, v）, even if there are 2n - 3 link faults.
Let G be a connected graph of order n. The diameter of a graph is the maximum distance between any two vertices of G. In this paper, we will give some bounds on the diameter of G in terms of eigenvalues of adjacency matrix and Laplacian matrix, respectively.
设G是阶为n边数为m的简单图,λ1,λ2,…,λn是G的邻接矩阵的特征值,μ1,μ2,…,μn是G的拉普拉斯矩阵的特征值.图G的能量定义为E(G)=sum from i=1 to n|λi|,拉普拉斯能量LE(G)=sum from i=1 to n|μi-(2m/n)|.利用代数和图论的方法,得到了k-正则图的最大和最小能量,以及最大、最小拉普拉斯能量,并刻划了能量取到最值时对应的图的结构.
