哈密顿图的无符号拉普拉斯谱半径条件

令A(G)=(a_(ij))_(n×n)是简单图G的邻接矩阵,其中若v_i-v_j,则a_(ij)=1,否则a_(ij)=0.设D(G)是度对角矩阵,其(i,i)位置是图G的顶点v_i的度.矩阵Q(G)=D(G)+A(G)表示无符号拉普拉斯矩阵.Q(G)的最大特征根称作图G的无符号拉普拉斯谱半径,用q(G)表示.Liu,Shiu and XueR.Liu,W.Shui,J.Xue,Sufficient spectral conditions on Hamiltonian and traceable graphs,Linear Algebra Appl.467(2015)254-255]指出:可以通过复杂的结构分析和排除更多的例外图,当q(G)≥2n-6+4/(n-1)时,则G是哈密顿的.作为论断的有力补充,给出了图是哈密顿图的一个稍弱的充分谱条件,并给出了详细的证明和例外图.

A Note on Signless Laplacian Spectral Condition of Hamiltonian Graphs

Let A(G) =(aij)nxn be the adjacency matrix of a simple graph G,where aij =1 if vi ～ vj,otherwise aij =0.Let D(G) be the diagonal matrix whose (i,i)-entry is the degree of the vertex vi of G.The matrix Q(G) =D(G) + A(G) is the signless Laplacian matrix.The signless Laplacian spectral radius of G is the largest eigenvalue of Q(G),denoted by q(G).Liu,Shiu and Xue R.Liu,W.Shui,J.Xue,Sufficient spectral conditions on Hamiltonian and traceable graphs,Linear Algebra Appl.467 (2015) 254-255] pointed out that by much more complicated analysis and excluding much more exceptional graphs,if q(G) ≥ 2n-6 + n/4-1,then G is Hamiltonian.As a supplementary,in this paper,we provide a weaker condition and give a detailed proof.