补图是树的图的距离拉普拉斯特征值

Distance Laplacian Eigenvalues of Graphs Whose Complements Are Trees

一个连通图的距离拉普拉斯矩阵定义为顶点传输度对角矩阵与距离矩阵的差,距离拉普拉斯矩阵的特征值称为这个图的距离拉普拉斯特征值.距离拉普拉斯伸展度定义为图的最大与次小距离拉普拉斯特征值的差.本文确定了补图的最大距离拉普拉斯特征值取得最小值和最大值的树及补图的次小距离拉普拉斯特征值取得最小值和最大值的树,也确定了补图的次大距离拉普拉斯特征值取得最小值的树,还确定了补图的距离拉普拉斯伸展度取得最小值和最大值的树.

The distance Laplacian eigenvalues of a connected graph are the eigenvalues of its distance Laplacian defined to be the difference between the diagonal matrix of vertex transmissions and the distance matrix.In this paper,we determine the unique trees for which the complements minimize(maximize,respectively)the largest distance Laplacian eigenvalue as well as the unique trees for which the complements minimize(maximize,respectively)the second smallest distance Laplacian eigenvalue.We also determine the unique trees for which the complements minimize the second largest distance Laplacian eigenvalue,and the unique trees for which the complements minimize(maximize,respectively)the distance Laplacian spread which is defined to be the difference between the largest and the second smallest distance Laplacian eigenvalues.