Research On Spectral Parameters Of Graphs And Related Problems

The theory of graph spectra mainly uses the spectra of some graph matrices to characterize the structure of a graph,such as the adjacency matrix,Laplacian matrix,signless Laplacian matrix and normalized Laplacian matrix.In this thesis,we mainly study the spectral parameters of graphs and related problems.The main content includes:1.Relations between energy,matching number and rank of graphs.We first establish upper bounds of graph energy in terms of matching number,and characterize the extremal graphs.We next determine the lower bound of the energy of graphs whose cycles are pairwise vertex-disjoint in terms of matching number,and characterize the extremal graph.In addition,we give a new lower bound for the energy of connected graphs whose ranks are greater than four,and characterize all connected graphs whose energy are less than the lower bound.2.On the normalized Laplacian spectrum involving graph operations and transformations.We first determine the normalized Laplacian spectrum,multiplicative degree-Kirchhoff indices and numbers of spanning trees of graphs obtained by the strong product of a path or a cycle and an edge,respectively.Next,we consider the influence on the normalized Laplacian spectrum by adding a triangle or a quadrangle to each edge of a graph.For graphs obtained by repeating one of the above two transformations r times,we characterize their normalized Laplacian spectrum,multiplicative degree-Kirchhoff indices and numbers of spanning trees in terms of these paprameters of original graphs.3.Relations between fractional matching and the signless Laplacian spectral radius of graphs.We first determine the lower bound of the fractional matching number of graphs in terms of signless Laplacian spectral radius,and characterize the extremal graphs.Moreover,we give some sufficient conditions for the existance of the fractional perfect matching of graphs.