单圈图与双圈图补图的A_(α)-谱半径

The A_(α) -spectral radius of the complement of unicyclic and bicyclic graphs with n vertices

设A(G)和D(G)分别表示n阶图G的邻接矩阵和度对角矩阵,对于任意实数α∈[0,1],图G的A_(α)-矩阵被定义为A_(α)(G)=αD(G)+(1-α)A(G),它是图的邻接矩阵和无符号拉普拉斯矩阵的共同推广,其最大特征根称为图G的A_(α)-谱半径.单圈图与双圈图补图的A_(α)-谱半径的上界被分别确定,相应的极图被完全刻画.

Let G be a simple graph with adjacency matrix A(G)and let D(G)be the diagonal matrix of the degrees of G.For every real α∈[0,1],the Aa-matrix of a graph G is defined as A_(α)(G)=αD(G)+(1-α)A(G),which is a common generalization of the adjacency matrix and the signless Laplacian matrix of a graph.The largest eigenvalueλ1(A_(α)(G))is called Aa-spectral radius of G.The upper bound on Aa-spectral radius of the complement of unicyclic and bicyclic graphs with n vertices is determined respectively,and the extremal graphs are characterized completely.