一类单圈图的谱

The Spectrum of some Unicyclic Graphs

设G是同一层的所有顶点的度数相等的k层单圈图,证明了G的邻接矩阵的特征值等于k阶非负对称三对角块矩阵的前主子矩阵的特征值,并且利用这个结论给出了单圈图邻接矩阵的最大特征值的一个上界:λ1(A(Gk))<max{max3≤j≤k{jγ-1+γj-1-1},γk-1+2},其中jλ=max{dx∶x∈G,dist(x,Cl)=k-j+1}.

Let G be an unicyclic graph of k levels such that in each level the vertices have the equal degrees. It is proved that the eigenvalues of the adjacency matrix equal to the eigenvalues of leading principal submatrices of nonnegtatire symmetric tridiagonal matrices of order k × k. By using these results, an upper bound for the largest eigenvalue of the adjacency matrix of any unicyclic graph is obtained： λ1（A（G^K））〈max{max3≤j≤k}√γj-1＋√γj-1-1},√rk-1＋2},where γj=max{dx：x∈L,dist（x,Cj）=k-j＋1}