超立方体的Laplace矩阵的谱

Spectra of Laplacian matrices of hypercubes

本文解决了超立方体的Laplace矩阵的谱问题.n维超立方体Qn的Laplace矩阵L(Qn)的谱specL(Qn)=[0 2 4…2n C0nC1nC2n…Cnn],其中2t(t=0,1,2,…,n)为L(Qn)的n+1个不同的特征值,二项式系数Ctn为特征值2t的重数.

For a simple graph G with vertex set {v1 ,v2 ,… ,vn } and adjacency matrix A（G）, the Laplacian matrix of G is L（G）= D（G）-A（G）, where D（G）= （dij ） is the diagonal matrix in which dii is the degree di of vi （Let d1 ≥d2≥…dn）. the Laplacian matrix is of great importance in graph theory, and its spectra is an very useful algebraic tool. The hypercubes appear often as models in computer science because of the useful properties of these graphs have. In this paper, the spectra of the Laplacian matrices of hypercubes was obtained. The spectra of the Laplacian matrix of n-hypercube are [0 2 4…2n Cn^0 Cn^1 Cn^2 … Cn^n],where the 2t（t= 0,1,2,…, n） are the eigenvalues of the Laplacian matrix of n-hypercube, and the binomial coefficients Cn are the multiplicities of the eigenvalues 2t.