Research On The Distance Spectrum And D_α-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 Dα(G)=αTr(G)+(1-α)D(G),0≤α≤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=Σvj∈V(G)dij(G).And the eigenvalue sets of D(G)and Dα(G)are called the distance spectrum and Dα-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 Dα(G)are the distance spectral radius and Dα-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 Dα-spectral radius(0≤α≤1/2),and as well,for the complements of trees,we depict the extremal graphs with the maximum and the minimum Dα-spectral radius(0≤α≤1),respectively.