具有最大Aα-特征值的符号完全图

On the largest Aα-eigenvalue of signed complete graphs

符号图Σ=(G,σ)由它的底图G=(V,E)与符号函数σ∶E→{-1,1}组成.设α∈[0,1],Belardo定义符号图Aα-矩阵为:A_(α)(Σ)=αD(G)+(1-α)A(Σ),其中A(Σ)是符号图Σ的邻接矩阵,D(G)是G的度对角矩阵.设K n表示n阶完全图,T表示树.设(K_(n),H^(-))表示负边导出子图是H的符号完全图.当0.5<α<1时,确定了符号图(K_(n),T^(-))的最大Aα-特征值达到最大时T的结构.

A signed graphΣ=(G,σ)consists of an underlying graph G=(V,E)and a sign functionσ∶E→{-1,1}.LetΣbe a signed graph,the matrix Aα(Σ)is defined by Belardo as follows:A_(α)(Σ)=αD(G)+(1-α)A(Σ),whereα∈[0,1],D(G)is the diagonal matrix of the vertex degrees of G and A(Σ)is the adjacency matrix ofΣ.Let K n denote the complete graph of order n and T be a tree.Let(K_(n),T^(-))be a signed complete graph whose negative edges induce a subgraph H.In this paper,we characterize the extremal signed graph with maximumλ1(A_(α)(Σ))among graphs of type(K_(n),T^(-)).