Research On The Distance Spectrum And D_?-spectrum Of Graphs

Spectral graph theory is an important embranchment of graph theory.Spectral graph theory does research on the properties of graphs mainly by adjacent matrix,Laplacian matrix and other matrices of the graphs.For a graph G,we define matrix D?(G)=?Tr(G)+(1-?)D(G),0???1,in which D(G)is the distance matrix of G with[D(G)]i,j=dij(G)while Tr(G)is a diagonal matrix with[Tr(G)]i,i=?vj?V(G)dij(G).And the eigenvalue sets of D(G)and D?(G)are called the distance spectrum and D?-spectrum of G,separately.The maximum module of eigenvalues of matrix M is called the spectral radius of M.Let G be a connected graph,then the maximum eigenvalues of D(G)and D?(G)are the distance spectral radius and D?-spectral radius of G,respectively.In this paper,for the complements of unicyclic graphs,we characterize the extremal graphs with the maximum distance spectral radius and also determine the extremal graph with the maximum the least distance eigenvalue where diameter of unicyclic graphs is three;further,we generalize the conclusions of the distance spectral radius we have drawn to D?-spectral radius(0???1/2),and as well,for the complements of trees,we depict the extremal graphs with the maximum and the minimum D?-spectral radius(0???1),respectively.