设G是一个简单图, A(G), Q(G)以及Q(G)分别为G的邻接矩阵, 无符号拉普拉斯矩阵以及距离无符号拉普拉斯矩阵, 其最大特征值分别称为G的谱半径, 无符号拉普拉斯谱半径以及距离无符号拉普拉斯谱半径. 如果图G中有一条包含G中所有顶点的路, 则称这条路为哈密顿路; 如果图G含有哈密顿路, 则称G为可迹图; 如果图G含有从任意一点出发的哈密顿路, 则称G从任意一点出发都是可迹的. 主要研究利用图G的谱半径, 无符号拉普拉斯谱半径, 以及距离无符号拉普拉斯谱半径, 分别给出图G从任意一点出发都是可迹的充分条件.

Let G be a simple graph, A(G), Q(G) and Q(G) are the adjacency matrix, the signless Laplacian matrix, and the distance signless Laplacian matrix of G, respectively. The largest eigenvalues of A(G), Q(G) and Q(G) are called the spectral radius, the signless Laplacian spectral radius and the distance signless Laplacian spectral radius of G, respectively. A path is called a Hamilton path if it contains all vertices of G. A graph is traceable if it contains a Hamilton path. A graph is traceable from every vertex if it contains a Hamilton path from every vertex. The main research of this paper is to give some sufficient conditions for a graph to be traceable from every vertex in terms of spectral radius, signless Laplacian spectral radius and distance signless Laplacian spectral radius of the graph, respectively.