X. Deng et al. proved Chvátal'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.
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.
