一类图的谱

设K_m是m阶完全图,将n+1个m阶完全图通过固定的方式连结,得到(mn+m)阶完全关联图H_n,K_m。在利用商矩阵及秩的相关结论后,给出了完全关联图H_n,K_m的邻接矩阵、拉普拉斯矩阵和无符号拉普拉斯矩阵的特征值,从而确定了完全关联图H_n,K_m的邻接谱、拉普拉斯谱和无符号拉普拉斯谱。同时,基于对Brualdi-Solheid谱半径问题的研究,并将这类谱半径问题推广到图的拉普拉斯谱半径和无符号拉普拉斯谱半径的研究中,给出了H_n,K_m(所有点数为N的完全关联图构成的集合,其中N=m(n+1))中邻接谱半径的上界,拉普拉斯谱和无符号拉普拉斯谱半径的上、下界;并刻画了H_n,K_m中邻接谱半径达到上界的极图,以及拉普拉斯谱和无符号拉普拉斯谱半径达到上、下界时的极图。

Spectra of a Class of Graphs

Let K_m is a complete graph of order m. By linking n+1 complete graphs of order m in a fixed way, a complete associated graph H_n,Km of order (mn + m) was obtained. The adjacency eigenvalues, Laplacian eigenvalues and signless Laplacian eigenvalues of H_n,Km were provided according to the relevant conclusions about quotient matrix and rank. So the adjacency spectrum, Laplacian spectrum and signless Laplacian spectrum of H_n,Km were determined. At the same time, based on the study of Brualdi-Solheid spectral radius, the research of this kind of spectral radius problem was extended to the study of Laplacian spectral radius and signless Laplacian spectral radius of graphs. The upper and lower bounds of the radius of adjacency spectrum, Laplacian spectrum and signless Laplacian spectrum of H_n,Km (the set of complete correlated graphs with points N where N=m(n + 1)) were provided. And the extremal graphs were described where the radius of adjacency spectrum attained the upper bound and Laplacian spectrum and signless Laplacian spectrum attained the upper or lower bounds.