The Signless Laplacian Spectral Radius With Some Graphs

In the theory of graph spectra, there are various matrices that are naturally associated with a graph, such as the adjacency matrix, the Laplacian matrix, the incidence matrix, the distance matrix and so on. One of the main purposes of graph spectra theory is to determine precisely how, or whether, properties of graphs are reflected in the algebraic properties (especially properties about eigenvalues, such as spectral radius, spectral uniqueness, spread, energy and so on) of such matrices. The signless Laplacian spectrum is more closely relative to the graph structures than other spectrum. In this paper, we use signless Laplacian spectrum to study the structures and properties of graph with given independence number, graph with given connectivity and graph with given diameter. As follows is the main results:(1)αK1(?)Kn_αis the unique graph with maximum signless Laplacian spectral radius in the graphs with given independence number.(2) Kr,n_r is the unique graph with maximum signless Laplacian spectral radius in the bipartite graphs with given connectivity.(3) Gd= G([d/2]·1,n-d,[d/2]·1) is the unique graph with maximum signless Laplacian spectral radius in gd.