On The Distance Laplacian And Distance Signless Laplacian Spectra Of Graphs

Several matrices have been introduced into the research of graph theory, such as incidence matrix, adja-cency matrix, Laplacian matrix, signless Laplacian matrix, and distance matrix etc. Linear algebra and matrix theory are applied to study such matrices defined by the structure of graphs, which helps people learn more about properties of graphs.Aouchiche and Hansen introduced distance Laplacian and distance signless Laplacian of a connected graph and studied their spectra. They proposed a series of conjectures about distance Laplacian spectrum and distance signless Laplacian spectrum. Some conjectures had been proved to be true. As we know, Laplacian matrix (or signless Laplacian matrix) have more information than adjacency matrix, because Laplacian matrix (or signless Laplacian matrix) shows the degrees of vertices directly in the diagonal entries. Similarly, distance Laplacian matrix (or distance signless Laplacian matrix) shows the sums of distance from one vertex to the others in the diagonal entries, but the distance matrix does not.In this paper, we study spectral radius of a distance Laplacian matrix and spectral radius of a distance signless Laplacian matrix in different classes of graphs.One part is for the distance Laplacian. Firstly, we prove a conjecture related to the multiplicity of the largest eigenvalue of distance Laplacian proposed by Aouchiche and Hansen. Secondly, by studying some graphs with a tree as its induced subgraph, we prove that its spectral radius must not be increasing when replace the subgraph with the same number of pendent vertices. And then, we determine the structure of trees with the smallest spectral radius among all trees with even diameters. Lastly, we compare the largest eigenvalues of distance Laplacian of unicyclic graphs with different girths.The other part is for the distance signless Laplacian. Firstly, by studying some graphs with a tree as its induced subgraph, we prove that its spectral radius must not be increasing when replace the subgraph with the same number of pendent vertices. Then, we discuss the eigenvector corresponding to the largest eigenvalue of the distance signless Laplacian of a class of unicyclic graphs, and we compare the largest eigenvalue of distance Laplacian of unicyclic graphs with different girths. At last, we characterize the structure of trees with the smallest spectral radius among all trees with even diameters.We list five conjectures in the summary, which are worth discussing deeply in the latter research.