Laplacian Spectral Radii Of Bipartite Graphs With Given Order And Size

The theory of graphs spectra is an active area in graph theory. There are extensive applications in lots of fields. In the theory of graphs spectra, there are many matrices are associated with graph for research the property of the graph, such as the adjacency matrix, the incidence matrix, the Laplacian matrix and the Q matrix and so on. In the process of study the theory of graphs spectra, the main idea is characterize the eigenvalues and other algebra properties of these matrices to react the construction property of the graphs. Among the above mentioned matrices of graphs, the most important are the adjacency matrix and the Laplacian matrix.Among the eigenvalues of the adjacency matrix and the Laplacian matrix, the most important is the largest eigenvalue, i.e., the spectral radius and the Laplacian spectral radius of the graph. Until now, the results about the bound of the spectral radius and the Laplacian spectral radius are much more ([1] gives the results about the bound of Laplacian spectral radius before 2005, [2], [3] and some other documents give more results about this). For some family of graphs which is given some parameters, the results about find the graph which has the maximum spectral radius of this family are less. However, the content about this aspect is regarded as more and more important in these several years, and the associated parameters are much more than before, such as the size, order, degree sequence, diameter, girth, maximum degree, minimum degree, pendant vertex number, cut edges and basic cycles and so on. In the process of study these problems, we can get the properties of different maximal graphs by research the different maximal graphs, and then study the whole structure and the correlative algebra properties.This paper is focus on the connected bipartite graphs with given order and size. It has been given the results about the problem of spectral radius of connected graphs with given order and size. In this paper, the focus is the correlative properties of the Laplacian spectra radius of this family graphs, and the graph which has the maximum Laplacian spectra radius of this family graphs is determined when n-1